Note that if (fn)nEIN and (gn)nEIN are both increasing (decreasing), then
in f) lim can be replaced by V (by 1\); see Proposition 1.8.19.
Having summarized the properties of real and extended-real functions, we
turn to the definition of a Riesz lattice. Essentially, the spaces we want to
describe are sublattices of lRx that are closed under addition and real scalar
multiplication. However, the fact that addition of two functions on X is only
defined when their sum is defined pointwise at every point of X creates a minor
inconvenience. We introduce some useful notations to overcome this.
Definition 2.1.7 -x -x Take f, 9 E lR and F, (} C lR . Then
(f -+ g) := {h E lRx I If x E X and f(x) + g(x) is defined, } then h(x) = f(x) + g(x).
(f ~ g) := (J -+ (-g»
If -+ 91 := {Ihl I h E (J -+ g)}
If ~ gl := {Ihl I hE (J ~ g)}
F -+ (}:= U (J -+ g) /E:F,9EQ
F ~ (}:= U (J ~ g) 0 /E:F,gEQ
Proposition 2.1.8 Let f, g, and h be extended-real functions on X. Then
a) (J -+ g) = (g -+ J) .
b) hE (J -+ g) iff 9 E (h ~ J) .
Proof. The truth of a) is evident. To prove b), suppose that h E (J -+ g) and
fix x in X. We distinguish three cases.
Case 1: If f(x) is real, then both f(x) + g(x) and h(x) - f(x) are defined,
h(x) = f(x) + g(x) , and g(x) = h(x) - f(x) .
2.1 Riesz Lattices 283
Case 2: Suppose that f(x) = 00. If h(x) = 00 also, then h(x) - f(x) is
undefined. If, on the other hand, h(x) < 00, then because h E (J -i- g) it must
be that f(x) + g(x) is not defined. In other words, g(x) = -00, and we have
g(x) = h(x) - f(x).
Case 3: Suppose finally that f(x) = -00. If h(x) = -00, then h(x) - f(x)
is not defined. If h(x) > -00, then hE (J -i- g) implies that f(x) + g(x) is
undefined, g(x) = 00, and g(x) = h(x) - f(x).
Thus in each case either h(x)- f(x) is undefined or else g(x) = h(x)- f(x).
Since x is arbitrary, it follows that 9 E (h -.:... 1) . The fact that 9 E (h -.:... 1) implies h E (J -i- g) can be proved by a similar
argument. 0
-x x Definition 2.1.9 A nonempty set .c c IR is called a Riesz lattice (in IR ,
or on X ) iff it satisfies the following three conditions. (RL 1) If f,g E.c, then (J -i- g) c.c. (RL 2) If f E.c and 0: E IR, then o:f E .c . (RL 3) If f, 9 E .c, then f 1\ 9 E.c and f V 9 E .c.
A Riesz lattice contained in IRx is called a real Riesz lattice. 0
IRX is trivially a Riesz lattice, as is the set {J E IRx I {J I- O} is finite}.
Every vector sublatticc of IRx is an example of a real Riesz lattice. For every
Riesz lattice .c in IRx , the set .c n IRx is a real Riesz lattice.
Why do we use this particular definition for function classes'? Any construc
tion of an integral starts with some convenient class of functions whose integrals
are known. The linearity properties of integrals only make sense if this class is
closed under addition and scalar multiplication. The class of functions whose
integrals have been defined is then expanded, possibly more than once. In this
book, one of the extensions is achieved via monotone approximation, and Axiom
(RL3) is used in the rnonotone--approximation process.
The easy proof of the next proposition is left to the reader.
Proposition 2.1.10 The following assertions hold for every Riesz lattice .c in IRx .
a) The zero function, 0, belongs to .c.
b) For every f in .c, - f belongs to .c.
c) For every f in .c, If I, rand f- belong to .c+.
d) If f and 9 belong to .c, then (J -.:... g) c .c .
284 2. Elementary Integration Theory
e) Given a finite family (f.).EI in £, n IRx , the sum E f. belongs to £' . • EI
f) Given a nonempty finite family (f.).EI in £', both V f. and "f. belong 'EI .EI
to £'. o
Many Riesz lattices have special order properties relative to the full space
IRx. We discuss these properties briefly.
Definition 2.1.11 A set :F c IRx is said to be:
a) conditionally a-completely embedded in IRx iff, given any nonempty
countable family (f.).EI in :F which is bounded in :F, both V f. and .EI
"f. belong to :F; .EI
b) a-completely embedded in IRx iff, given any nonempty countable family
(f.).EI in :F, both V f. and "f. belong to :F. 0 .EI .EI
Every (conditionally) a-completely embedded lattice in IRx is itself a (con
ditionally) a-complete lattice, with order relation induced by the order relation
on IRx , but the converse is ~ot true. Obviously every set that is a-completely
embedded in IRx is also conditionally a-completely embedded.
When the set in question is a Riesz lattice, the test for a-complete em
bedding or conditionally a-complete embedding is easier. One need not test
all countable families. It is enough, for instance, to test positive increasing se
quences. More precisely, we have the following two propositions.
Proposition 2.1.12 Let £, be a Riesz lattice in IRx . Then the following are
equivalent.
a) £, is conditionally a-completely embedded in IRx.
b) Given any sequence (fn)nEIN in £, which is bounded in £', both V fn nEIN
and " f n belong to £'. nEIN
c) Given any sequence (fn)nEIN in £, which is decreasing and bounded below
in £', the function " fn belongs to £'. nEIN
d) Given any sequence (fn)nEIN in £,+ which is increasing and bounded above
in £', the function V fn belongs to £'. nEIN
2.1 Riesz Lattices 285
Proof. a) - b) and b) =:} c) are trivial (and hold for arbitrary subsets .c of
IRx ).
c) =:} d). If (fn)nEIN is a sequence in .c+ which is increasing and bounded
above in .c, then (- in)nEIN is a sequence in .c which is decreasing and bounded
below in .c. Since
nEIN
it follows that V in belongs to .c. nEIN
nEIN
d) =:} b). Let (fn)nEIN be a sequence in .c which is bounded in .c. For
each n E IN , define
if defined
otherwise.
The sequence (hn)nEIN increases and each hn is positive. Since
we see that every hn is in .c and therefore in .c+. To verify that the sequence
(hn)nEIN is bounded above in .c, note that if g' and g" are elements of .c which
bound the original sequence (fn)nEIN from above and below, respectively, then
the function 19'1 + 19"1 is defined, belongs to .c and bounds (hn)nEIN from
above. Hypothesis d) implies that V hn belongs to .c. Now nEIN
V hn E / V ~ il) , nEIN \ nEIN
so
V in E / V hn -+- it), nEIN \ nEIN
by Proposition 8 b). We conclude that V in must belong to .c. nEIN
To conclude that V in belongs to .c, we argue as follows. Since (fn)nEIN nEIN
is bounded in .c, so is (- in)nEIN. The last argument shows that V (-in) nEIN
belongs to .c. Now recall that 1\ in = - V (-in). o nEIN nEIN
286 2. Elementary Integration Theory
Proposition 2.1.13 The following conditions on a Riesz lattice .c in lRx
are equivalent.
a} .c is a-completely embedded in lRx .
b} Given any sequence (fn)nEJN in .c, both V fn and 1\ fn belong to .c. nEJN nEJN
c) Given any decreasing sequence (fn)nEJN in .c, the function 1\ fn belongs nEJN
to .c.
d) Given any increasing sequence (fn)nEJN in .c+, the function V fn be-nEJN
longs to .c.
Proof. a) - b) and b) ::::} c) are trivial.
c) ::::} d). For every increasing sequence (fn)nEJN in .c+, the sequence
(- fn)nEJN is a decreasing sequence in .c and V fn = - 1\ (- fn) . nEJN nEJN
d) ::::} b). From d) and the preceding proposition it follows that .c is con-
ditionally a-completely embedded in lRx. It therefore suffices to show that
every sequence in .c is bounded in .c. Let (fn)nEJN be a sequence in .c. Then
is an increasing sequence in .c+, and d) ensures that its supremum, V Ifni, nEJN
belongs to .c. Since
-V Ifni::; fm ::; V Ifni nEJN nEJN
for every m in 1N the proof is complete. o
It is useful to single out those subsets of X on which some element of the
Riesz lattice takes infinite values.
Definition 2.1.14 Let.c be a Riesz lattice in lRx. A set A C X is called
.c-ezceptional iff there is a function f belonging to .c such that
f(x) = 00 for every x in A.
-x A function f E lR is called .c-ezceptional iff {J =f. O} is an .c-exceptional
set.
lJ1(.c) shall denote the collection of all .c-exceptional subsets of X, and
N(C) the collection of all .c-exceptional functions on X. 0
2.1 Riesz Lattices 287
Thus
1)1(£) = {A c X I A C {g = oo} for some 9 E £}
and
-x N(£) = {f E IR I {f =f. O} C {g = oo} for some 9 E£}.
If f belongs to £, then each of the sets {f = oo}, {f = -oo}, and {If I = oo}
is £-exceptional.
If £ = IRx , then 1)1(£) = I.l3(X) and N(£) = £. If £ is a real Riesz
lattice, then 1)1(£) = {0} and N(£) = {O} . There are many ways of characterizing £-exceptional sets and functions,
several of which are presented in the next two propositions.
Proposition 2.1.15 For every Riesz lattice £ in IRx, the following condi
tions on a subset A of X are equivalent.
a) A E 1)1(£) .
b) ooeA E £.
c) A C B for some B E 1)1(£) .
d) A C {f =f. O} for some f E N(£) .
Proof. a) => b). By hypothesis, there is an f in £ such that f(x) = 00 for
every x in A. For this f,
so ooe A must belong to £.
b) => a). This is obvious.
a) => c). Take B := A. c) => d). Take f := eB.
ooeA E (J ...:.. J)
d) => a). By hypothesis, A C {f =f. O} c {g = oo} for some f in IRx and
some 9 in £.. Thus A belongs to 1)1(£). 0
Proposition 2.1.16 For every Riesz lattice £ in IRx, the following condi
tions on a function f E IRx are equivalent.
a) f E N(£).
288 2. Elementary Integration Theory
b) If I E N(£) .
c) oof E £.
d) {J 1= O} C {g 1= O} for some 9 E N(£) .
e) {J 1= O} C A for some A E 91(£) .
Proof. a) =} b). After all, {If I 1= O} = {J 1= O}.
b) =} c). Put A := {If I 1= O}. By the previous proposition, the function
ooe A belongs to £. Since
oof must also belong to £. c) =} a). If oof E £, then loofl E £ and so f E N(£) .
a) =} d). Put 9 := f· d) =} e). Put A:= {g 1= O} and use Proposition 15 d) =} a).
e) =} a). By hypothesis, {f 1= O} cAe {g = oo} for some 9 in £. Thus
{J 1= O} is in 91(£); that is, f belongs to N(£). 0
The characterizations
N(£) = {J E ffi.x I oof E .c} and 91(£) = {A c X I ooeA E £}
are especially useful.
Proposition 2.1.17 Let £ be a Riesz lattice in ffi.x. Then I.P(A) C 91(£)
for each A E 91(£), and the union of every finite family in 91(£) belongs to
91(£) .
Proof. The first claim merely restates Proposition 15 c) =} a) . Let (A'),El be
a finite family from 91(£). Since
ooe U A, = V ooe A, ,
'EJ 'El
we conclude that ooe U A, belongs to £ and that U A, belongs to 91(£). 0 ,EJ 'EI
Corollary 2.1.18 If f and 9 are arbitrary elements of a Riesz lattice £ in -x ffi. ,then the set
{x E X I f(x) + g(x) is not defined}
is £-exceptional.
2.1 Riesz Lattices 289
Proof. The set in question is a subset of the set
{J = oo} U {g = oo} . o
Definition 2.1.19 Let C be a Riesz lattice in IRx. A property P depending
on elements of X is said to hold C-almost everywhere (or simply C-a.e.) iff
the set
{x E X I P(x) is false or P(x) is not defined}
is an C-exceptional set.
If P holds C-almost everywhere, then we write
P C-a.e.
provided that P(x) is defined for every x in X, and we write
P(x) C-a.e.
in case P(x) is not necessarily defined for every x in X. o
Example 2.1.20 Operations and relations on the set IRX were defined point
wise. Thus properties offunctions in IRx are properties which refer to points of X , and it is meaningful to assert that such a property holds £-almost every
where. Let f and 9 be extended-real functions on the set X, let (fn)nEIN be a sequence from IRx , and let £ be a Riesz lattice in IRx. Then
f = 9 £-a.e. {::=} {J =F g} E 1)1(£) ,
f ~ 9 £-a.e. {::=} {J > g} E 1)1(£) ,
f(x) = lim fn(x) C-a.e. {::=} n-+oo
{ (fn(X))nEIN does not converge
{::=} x EX or lim fn(x) =F f(x)
n-+oo
} E 'It(C).
Moreover, every function in £ is C-a.e. finite, and f(x) + g(x) is defined £
a.e. for all functions f and 9 in £. Every C-exceptional function equals zero
C-a.e., and conversely. If h is a function in (f + g) , where f and 9 belong
to £, then h(x) = f(x) + g(x) £-a.e. 0
290 2. Elementary Integration Theory
Proposition 2.1.21 Let C be a Riesz lattice in IRx .
a) Every C-exceptional function belongs to C. In other words, N{C) c C.
b) If A is an C-exceptional subset of X, then feA belongs to N{C) for
every extended-real function f on X, and feX\A belongs to C if f
belongs to C.
c) Every function which is C-almost everywhere equal to a function belonging
to C must itself belong to C.
d) If f and g belong to C and if
h{x) = f{x) + g{x) C-a.e.
then h belongs to C.
Proof a) If f belongs to N{C) , then oof belongs to C by Proposition 16
a) => c) . Since
f E (oof ~ oof)
f must also belong to C. -x
b) Take A E I)'1{C) and f E IR . Since
{feA =f. O} c A,
feA belongs to N{C) by Proposition 16 e) => a). By a), feA belongs to C.
The last claim now follows, since
-x c) Suppose that f = g C-a.e., for some f E C, g E IR . Put
A:={f=f.g}.
By a) and b), the functions feX\A and geA belong to C. Moreover, their sum
is defined and equals g. Hence g belongs to C.
d) Let
A := {x E X I f{x) + g{x) is not defined}
and let
h' := feX\A + geX\A .
Then A is C-exceptional by Corollary 18, so h' = h C-a.e. In view of b), h'
belongs to C. By c), h belongs to C. 0
2.1 Riesz Lattices 291
Proposition 2.1.22 If.c is a Riesz lattice in IRx , then .c-a.e. equality of
functions is an equivalence relation on the set IRx .
Proof. Reflexivity and symmetry are trivial. To prove transitivity, take f, 9, hE -x IR such that f = 9 .c-a.e. and 9 = h .c-a.e. The inclusion
{J :;': h} c {J :;': 9} U {9 :;': h}
then implies that {J:;': h} is .c-exceptional (Proposition 17). o
The proof of the next observation is left to the reader as an exercise.
Proposition 2.1.23 Let.c be a Riesz lattice in IRx and take f, 9, hE IRx .
a) If f ::5 9, then f ::5 9 .c-a.e.
b) f = 9 .c-a.e. iff f ::5 9 .c-a.e. and 9::5 f .c-a.e.
c) If f ::5 9 .c-a.e. and 9::5 h .c-a.e., then f ::5 h .c-a.e.
-x In other words, ::5 .c-a.e. is a preorder on IR . o
Proposition 2.1.24 Let.c be a Riesz lattice in IRx . Suppose that fl> hand
91, 92 are functions in 1R x such that
fi = 9i .c-a.e.
for i = 1, 2, and let h be an extended-real function on X. Then
a) O'.fl = 0'.91 .c-a. e. for every 0'. E IR.
b) f1l2 = 9192 .c-a.e.
c) If h(x) = fl(x) + h(x) .c-a.e., then
h(x) = gl(x) + 92(X) .c-a.e.
Now suppose that (f')'EI and (g')'EI are finite families in IRx such that
f, = gL .c-a.e.
for every £ in I. Then
d) V fL = V 9, .c-a.e. LEI LEI
292 2. Elementary Integration Theory
e) A It = A gt £-a.e. tEl tEl
Proof The key here is once again the closure of 91(£) under subsets and finite
unions. Thus the proposition is a consequence of the following five inclusions,
one for each assertion in the proposition:
a) {all =f. agd C UI =f. gd
b) {1I12 =f. glg2} C {II =f. gd U U2 =f. g2}
c) {x E X I gl(X) + g2(X) is undefined or =f. h(x)}
C {x E X I II (x) + l2(x) is undefined or =f. h(x)} U UI =f. gd U {h =f. g2}
d) {V/t=f.Vgt}cUUt=f.gJ tEl tEl tEl
e) {A It =f. A gt} C UUt =f. gJ. o tEl tEl tEl
2.2 Daniell Spaces 293
2.2 Daniell Spaces
Daniel spaces are the basis for the construction of an integral. This section
presents the necessary definitions and elementary properties of such spaces.
Definition 2.2.1 Take Fe IRx . A junctional on F is a real-valued func
tion with domain F. A functional £ on F is said to be:
a) additive if given f, g, hE F,
£(h) = £(f) + £(g)
whenever h E (f -+- g) ;
b) homogeneous if given f E F and a E IR,
£(af) = a£(f)
whenever af E F;
c) linear if it is both additive and homogeneous;
d) positive if £(f) 2: 0 for every f in F+;
e) increasing if it increases relative to the order relations on F and IR. o
Notice that if F is a real Riesz lattice, then the preceding definitions are
consistent with Definition 1.5.1.
Although no occasion for confusion should arise, a word of caution is proba
bly in order. Take Fe IRx . If £ : F ~ IR is viewed as an element of IRF , that
is, as a real-valued function on F, then £ is positive iff £(f) 2: 0 for every
f E F, according to Definitions 2.1.4 and 2.1.1. However, with £ : F -t IR
viewed as a real functional on F, £ is positive iff £(f) 2: 0 for every f E F+ ,
by Definition 1 d).
Proposition 2.2.2 Let £ be a functional on a Riesz lattice C. in IRx .
a) Suppose that £ is additive. Then £(0) = 0 and £( - f) = -£(f) for every
f E c.. If f, 9 E C. and hE (f .:.- g) , then
£(h) = £(f) -£(g) .
294 2. Elementary Integration Theory
b) If £ is additive and (f')'EI is a finite family in £. n IRx , then
£ ( 2: f,) = 2: £(f,) . 'EI ,EI
If £ is also homogeneous and (a')'EI E IR I , then
£ ( 2: ad,) = 2: a/(f.) . ,EI ,EI
c) If £ is additive, then £ is positive iff £ is increasing.
d) If £ is increasing, then for every nonempty family (f')'EI in £. for which
1\ f, E £. ,EI
£ (/\ fL) :S inf £(fL) , ,EI LEI
and for every nonempty family (fL),El in £. for which V f, E £. LEI
£ (v f,) ~ sup £(f,). LEI LEI
e) If £ is homogeneous and increasing, then
1£(f)1 :S £(Ifl)
for every f in £..
Proof. The proofs of assertions a), b), and d) are left to the reader.
c) In view of a), £ additive and £ increasing certainly imply £ positive.
Conversely, assume that £ is additive and positive. Given f and 9 in £. with
f :S g, define
{ g(x) - f(x)
h : X ---+ IR, X t-----+ o
if defined
otherwise.
Evidently, h is a positive function belonging to (g"':" f) . It follows by a) that
O:S f(h) = £(g) - f(f).
Thus £(f) :S f(g) and £ increases.
e) From -If I :S f :S If I we conclude that
-£(Ifl) :S £(f) :S £(Ifl) . o
2.2 Daniell Spaces 295
Proposition 2.2.3 Let £ be a positive linear functional on a Riesz lattice C -x -x in IR . Take f, 9 E C and hEIR . Let A be an C-exceptional set.
a) If f belongs to N(C) , then £(J) = £(Ifl) = o.
b) £(JeA) = 0, £(JeX\A) = £(J) .
c) If f :::; g C-a.e., then £(J) :::; £(g) .
d) If h = f C-a.e., then h belongs to C and £(h) = £(J).
e) If h(x) = f(x) + g(x) C-a.e., then h belongs to C and
£(h) = £(J) + £(g) .
Proof. a) If f belongs to N(C) , then the functions f, If I , oof, and loofl all belong to C (Propositions 2.1.16, 2.1.21). Now
loofl + loofl = loofl ,
so
£(Ioofl) + £(Ioofl) = £(joofD .
Since £ takes only real values, we conclude that £(Ioofl) = o. The inequality
o ::; If I ::; loofl
yields
o ::; £(Ifl) ::; £(Ioofl) = 0
(Proposition 2 c)) and therefore £(Ifl) = o. By Proposition 2 e), £(J) = o. b) follows from a) and Proposition 2.1.21 b).
c) Put B := {J > g}. By hypothesis, B is C-exceptional, and
By Proposition 2 c), £ is increasing. Using b), we have
d) is a consequence of c) and Propositions 2.1.21 c) and 2.1.23 b).
e) That h belongs to C was established in Proposition 2.1.21 d). Put
296 2. Elementary Integration Theory
B := {x E X I f(x) + g(x) is not defined}
and note that B is .c-exceptional. Using b) and d), we have
f(h) = f(feX\B + geX\B) =
= f(feX\B) + f(geX\B) =
= f(f) + f(g) . o
It is now clear why the function values ±oo do not disturb the real
valuedness of positive linear functionals on Riesz lattices. For a Riesz lattice
.c only admits positive linear functionals which ignore the sets on which func
tions from .c take infinite values.
The preceding considerations enable us to make the following definition.
Definition 2.2.4 Let f be a positive linear functional on the Riesz lattice .c,
and take f, 9 E .c. Then
f(f ± g) := f((I ± g}) := f(h) ,
f(lf ± gl) := f(l(I ± g}1) := f(lhl) ,
where h is an arbitrary element of (I ± g) . o
Note, however, that the expression f(f + g) need not have meaning.
Our construction of integrals starts with positive linear functionals satisfy
ing a rather weak continuity or convergence condition. The condition in question
is described in the next definition.
Definition 2.2.5 A functional f on a set :F c IRx is said to be nullcontin
uous iff
lim f(fn) = 0 n ..... oo
for every decreasing sequence (fn)nEIN in :F for which
/\ fn = o. nEIN
A Daniell space is a triple (X,.c, f), where .c is a Riesz lattice in IRx
and f is a positive, linear, nullcontinuous functional on .c. o
2.2 Daniell Spaces 297
The name "Daniell space" is chosen in recognition of the work of P.J.
Daniell, who in 1918 constructed an integral starting with a positive linear
nullcontinuous functional on a real Riesz lattice.
It is important that in Definition 5 1\ is taken in IRx and not in F. Therefore, if ( is a positive linear functional on a real Riesz lattice £', null
continuity and O'-continuity of ( (see Definition 1.7.1) do not coincide! It is
immediate that O'-continuity of ( implies nullcontinuity, but the converse need
not hold as the following example shows. (The converse is true, however, if £,
is conditionally 0'- completely embedded in IRx .)
Example 2.2.6 Put £, := C([O, 1]) and define ( by ((f) := f(O) for every
f E£'. Then ( is a positive linear functional on £, which is trivially nullcon
tinuous. Now consider the sequence (fn)nEIN from £,+, defined by
fn(x) := 1/\ nx (x E [0,1]) .
Then V fn = e[O,!] where V is taken in C([O, 1]). But ((fn) = 0 for every nEIN
n E IN while (e[O,Ij) = 1. Thus ( is not O'-continuous (see Proposition 1.7.3.
a) => c)). 0
Proposition 2.2.7 Let ( be a positive linear functional on a Riesz lattice £,
in IRx. Then the following are equivalent.
a) ( is nullcontinuous.
b) inf ((fn) = 0 for every decreasing sequence (fn)nEIN from £, which satnEIN
isfies 1\ fn = O. nEIN
c) For every increasing sequence (fn)nEIN from £', if V fn belongs to £', nEIN
then
( (V fn) = sup ((fn) . nEIN nEIN
d) For every decreasing sequence (gn)nEIN from £', if 1\ gn belongs to £', nEIN
then
( (1\ gn) = inf (gn) . nEIN
nEIN
298 2. Elementary Integration Theory
Proof. a) - b) follows easily from the fact that f is increasing (Proposition
2 c)).
b) ::::} c). Let (In)nEIN be an increasing sequence from £, for which
1:= V In nEIN
belongs to £'. Given n E 1N , define
and
hn : X --; lR , ..----r - X' ----'- { 10 (x) - In(x) if defined
otherwise
h~ : X --; lR, if hn(x) < 00
if hn(x) = 00.
Then hn E (f ...:.. in) . Hence every hn belongs to £'. Thus for each n,
h~ = hn £'-a.e. ,
h~ belongs to £', and f(h~) = f(hn) . The sequences (hn)nEIN and (h~)nEIN are both decreasing and
c) ::::} d). Use the formula 1\ gn = - V (-gn). nEIN nEIN
d) ::::} b) is trivial. o
We now investigate a stronger form of functional convergence, suitable when
topological compatibility is desired.
2.2 Daniell Spaces 299
Definition 2.2.8 Let £ be an increasing functional on a subset :F of lRx .
A sequence (fn)nEIN from :F is called an i-sequence if (fn)nEIN is monotone
and (£(fn))nEIN is bounded in lR. The triple (X,:F, £) is said to be closed iff,
for every i-sequence (fn)nEIN from :F, the function lim fn belongs to :F and n-+oo
£ ( lim fn) = lim £(fn) . n--+oo n--+oo
o
-x Notice that for each monotone sequence (fn)nEIN from lR , lim fn exists.
n-+oo
Moreover, if £ is an increasing functional on some subset :F of lRx , then for
every i-sequence (fn)nEIN from :F, the sequence (£(fn))nEIN converges in lR.
For then (£(fn))nEIN is a monotone sequence of real numbers bounded in lR.
The closure property just defined is an axiomatization of the theorem from
classical integration theory known as the Beppo Levi Theorem (1906). Daniell
spaces that are also closed are the primary objects, from an abstract point of
view, to be studied in this part of the book.
Given a Riesz lattice C in lRx and a positive linear functional £ on C,
if the triple (X, C, £) happens to be closed, then it is immediate from the
definitions that the functional f is nullcontinuous. Thus to show that a given
triple is a closed Daniell space, it is redundant to verify the nullcontinuity.
Proposition 2.2.9 Let f be a positive linear functional on a Riesz lattice C
in lR x . Then the following are equivalent:
a) The triple (X, C, f) is closed: that is, (X, C, £) is a closed Daniell space.
b) For every increasing i-sequence (fn)nEIN from C, the function V fn nEIN
belongs to C and
£ (V fn) = sup £(fn) . nEIN nEIN
c) For every increasing i-sequence (fn)nEIN of positive functions from C,
the function V fn belongs to C and nEIN
£ (V fn) = sup f(fn) . nEIN nEIN
d) For every decreasing i-sequence (fn)nEIN from C, the function 1\ fn nEIN
belongs to C and
300 2. Elementary Integration Theory
£ (1\ In) = inf £(fn) . nEIN
nEIN
Proof a) '* b) '* c) is evident.
c) '* d). Let (fn)nEIN be a decreasing £-sequence from 1:-. For each n E IN ,
define
Now apply c) to the sequence (hn)nEIN, noting that
if defined
otherwise.
d) '* a). We have to show that lim In E I:- and that £ (lim In) = lim £(fn) n-+oo n-+oo n--+oo
whenever (fn)nEIN is an increasing £-sequence from 1:-. This follows easily using
the formula V In = - /\ (-In). 0 nEIN nEIN
Example 2.2.10 Let L: denote the positive linear functional
£1(X) ---+lR, II----t LI(x). xEX
We show that (X, £1 (X), L:) is a closed Daniell space.
Let (fn)nEIN be an increasing sequence from £1 (X)+ such that
a:= sup Lln(X) < 00. nEIN xEX
Put 1:= V In. Then, using Proposition 1.1.17 g), nEIN
L I(x) = L (sup In(X)) = sup (L In(x)) :::; a xEA xEA nEIN nEIN xEA
for every finite subset A of X. It follows that IE £1(X) and L: I(x) :::; a. xEX
Since I ~ In for every n E IN, L: I(x) ~ a. Hence xEX
LI(x)=a, xEX
and by Proposition 9 c) '* a), (X, £1(X), L:) is a closed Daniell space (which
merely formulates the fact that the normed vector lattice £1(X) is strongly
complete). 0
2.2 Daniell Spaces 301
The exceptional sets and functions of closed Daniell spaces have special
properties, as does the behaviour of the functional relative to a.e.-conditions.
The next propositions describe these special properties.
Proposition 2.2.11 If (X, C, l) is a closed Daniell space, then the following
conditions on a function f in lRx are equivalent.
a) f E N(C) .
b) If I E C and l(lfl) = o.
Proof. a) => b) for any positive linear functional i on any Riesz lattice C
(Proposition 3 a)).
b) => a). By Proposition 2.1.16 c) => a) , b) => a), it suffices to show that
oolfl belongs to C. Indeed, the sequence (nlfl)nEIN is an increasing i-sequence from C with supremum oolfl. Since the triple (X, C, l) is closed, we conclude
that oolfl belongs to C. 0
Corollary 2.2.12 If (X, c, i) is a closed Daniell space, then the following
conditions on a subset A of X are equivalent.
a) A E ')1(C) .
o
Proposition 2.2.13 Let (X, C, l) be a closed Daniell space, and let f and
9 be functions in C such that
f $ 9 and i(f) = i(g) .
Then
f = 9 C-a.e.
-x Moreover, if h E lR with f $ h $ g, then hE C and
Proof. Define
f': X ---t JR,
l(h) = i(f) = £(g) .
{ g(x) - f(x)
X t---+ o if defined
otherwise.
302 2. Elementary Integration Theory
Evidently, f' E (g -.:. f) . Hence h' is in £+, and
£(f') = £(g) - £(f) = o.
It follows (by Proposition 11) that f' is an £-exceptional function, and so
U # h} C U' # O} E 1)1(£).
In other words, f = 9 £-a.e.
If hE IRx with f ::; h ::; g, then the inclusion
U#h}CU#g}
shows that f = h £-a.e. An appeal to Proposition 3 d) now completes the
~~ 0
Finite unions of £-exceptional sets are still £-exceptional. In the case of
a closed Daniell space, "finite" can be replaced by "countable".
Proposition 2.2.14 If (X, £, £) is a closed Daniell space and (A,),E/ is a
countable family from 1)1(£), then U A, also belongs to 1)1(£). ,EI
Proof. If suffices to treat the case I = IN. So let (An)nEIN be a sequence from
1)1(£) . The sequence (fn)nEIN, where
for all n E IN, is an i-sequence from £: it is increasing and £(fn) = 0 for
every n (Proposition 2.1.17 and Corollary 12). Because the triple (X, £, £) is
closed, we conclude that the function V fn belongs to £ and nEIN
£ (V fn) = sup £(fn) = O. nEIN nEIN
But V fn is just the characteristic function of the set U An. In view of nEIN nEIN
Corollary 12 b) =} a), U An belongs to 1)1(£). 0 nEIN
Corollary 2.2.15 If (X, £, £) is a closed Daniell space, then N(£) is a
completely embedded in IRx . In other words, the supremum and the infimum of
a nonempty countable family of £-exceptional functions are also £-exceptional.
2.2 Daniell Spaces 303
Proof. It suffices to show that the supremum of every increasing sequence of
positive £-exceptional functions is itself £-exceptional (Proposition 2.1.13 d)
=> a)). If (fn)nEJN is such a sequence, then
{V fn -# o} = U {In -# o} nEJN nEJN
and V fn' in view of Proposition 14 and Proposition 2.1.16 e) => a), must nEJN
belong to N(£). 0
Corollary 2.2.16 Suppose that (X, £, C) is a closed Daniell space and that
(f')'EI and (g,),o are countable families from lRx such that
f, ::; g, £ -a. e.
for every ~ in I. Then
1\ f, ::; 1\ g, £ -a. e. ,EI 'EI
and
V f, ::; V g, £-a.e. o 'EI 'EI
Having discussed the properties of the exceptional sets and exceptional func
tions associated with closed Daniell spaces, we turn to the more significant ques
tion of the compatibility of closed Daniell spaces and their functionals with the
topology of lRx . The remainder of this section describes both how the Riesz
lattice £ is embedded in lR x and the accompanying convergence behaviour of
the functional £.
Theorem 2.2.17 Let (X, £, £) be a closed Daniell space and (f')'EI a non
empty, countable family from £.
a) If the family (f,) ,E I is directed up relative to the relation ::; £ -a. e., then
the two conditions
sup£(f,) < 00
'EI and
are equivalent and each implies that
304 2. Elementary Integration Theory
b) If the family (fL)LEl is directed down relative to the relation < .c-a. e.,
then the two conditions
inf £(ft) > -00 tEl
and
A. ft E .c LEI
are equivalent and each implies that
£ (A. fL) = inf £(ft) . LEI
LEI
Proof a) Let cp : IN -+ I be surjective. Construct an increasing sequence
(gn)nEIN in .c by setting
Clearly,
gn:= V f<p(m).
m~n
V gn = V fL. nEIN tEl
Moreover, as we see by complete induction, for each n III N there is an index
(n in I such that
and thus, by Proposition 3 c), such that
If
sup £(f.) < 00 LEI
then
In this case (gn)nEIN is an increasing £-sequence from .c. Since the triple
(X,.c, £) is closed, we conclude that V gn belongs to .c and thus V fL nEIN LEI
2.2 Daniell Spaces 305
belongs to C. Conversely, if we assume that V f, belongs to C, then the .el
monotonicity of f yields
sup f(J,) ::; f (V f,) < 00 . ,el ,el
Finally, if either of the two conditions holds, then so does the other and
f (V f,) = f (V gn) = sup f(gn) ::; sup f(J,J ::; ,el nelN ,el ,el
::; sup f(J,) ::; f (V f,) , ,el ,el
from which it follows that
f (~f,) = ~~ff(J,). b) can be proved analogously. o
Proposition 2.2.18 Let (X, C, f) be a closed Daniell space and let (J,),el
be a nonempty, countable family from C.
a) If there is a function 9 in C such that f, ::; 9 C-a.e. for every L in I,
then V f, belongs to C. ,el
b) If there is a function Y in C such that f, ~ 9 C-a.e. for every L in I,
then /\ f, belongs to C. ,el
Proof. a) Let t.p : IN -+ I be surjective. As in the proof of the preceding
theorem we construct an increasing sequence (Yn)nelN in C by putting
gn:= V fcp(m)'
m:Sn
The hypothesis ensures, that for each n E IN ,
gn::; 9 C-a.e.
Hence
Once again, (Yn)nelN is an increasing f-sequence from C. We conclude that
V f, = V Yn E C. ,el nelN
b) can be proved analogously. 0
306 2. Elementary Integration Theory
Corollary 2.2.19 For every closed Daniell space (X,.c, €), the Riesz lattice
.c is conditionally a-completely embedded in lRx . 0
Corollary 2.2.20 Let (X,.c, €) a closed Daniell space and Un)nEJN a se
quence from .c. If there is a function 9 in .c with
ifni :S 9 .c-a.e.
for every n, then we have the following.
a) lim sup fn belongs to .c and €(lim sup fn) 2: lim sup €( Un) . n-->oo n-->oo n-->oo
b) lim inf fn belongs to .c and €(lim inf fn) :S lim inf fUn) . n-+oo n-+oo n-+oo
Proof. a) Given n in IN, we put
gn:= V 1m. m2::n
Proposition 18 a) ensures that each gn belongs to .c. Moreover, in view of
We introduce CJ.. and fJ.. in complete analogy with Ct and ft.
Definition 2.3.3 Let (X, C, f) be a Daniell space. Then
CJ.. := { 1\ fn I (fn)nElN is a decreasing sequence from C n RX} nElN
and, for f E cJ.. ,
fJ..(!):= inf{f(g) I 9 E CnRx, g;::: J}. o
2.3 The Closure of a Daniell Space 311
The properties of (.CJ., fJ.) can easily be derived from those of (.ct, if) by
means of the following proposition.
Proposition 2.3.4 Let (X,.c, f) be a Daniell space and take f E lRx . Then
f belongs to .cJ. iff - f belongs to .ct and in this case
fJ.(f) = -ft( - f).
Proof The sequence (fn)nEIN from lRx is decreasing, after all, iff the sequence
(- fn)nEIN is increasing. Moreover /\ fn = - V (- fn) . nEIN nEIN
If f belongs to .cJ., then
fJ.(f) = inf{f(g) I 9 E .cnlRx , 9 ~ J} =
= - sup{ -f(g) I 9 E .c n lRx , 9 ~ J} =
= -sup{f(-g) I-g E .cnlRx , -g:::; -J} =
o
Corollary 2.3.5 Let (X,.c, e) be a Daniell space. Then the following hold.
a) .c n lRx c.cJ. and eJ.(f) = e(f) for every f E .c n IRX •
b) For every f E .cJ., eJ.(f) < 00.
c) The conditions f,g E.cJ. and f :::; 9 imply that eJ.(f) :::; eJ.(g).
d) If f and 9 belong to .cJ., then f + 9 is defined and belongs to .cJ., and
e) If f belongs to .cJ., then for every a E lR+, af belongs to .cJ. and
f) If f and 9 belong to .cJ., then so do f V g and f 1\ g.
g) If (fn)nEIN is a decreasing sequence from .cJ., then /\ fn belongs to .cJ. nEIN
and
o
312 2. Elementary Integration Theory
Proposition 2.3.6 Let (X, £, £) be a Daniell space. Take f E £i and 9 E
£t . If f ~ g, then
Proof. We have - f E £t , 9 + (-f) E £t and 9 + (-f) ~ O. Thus
o
Definition 2.3.7 Let (X, £, £) be a Daniell space. Suppose that f E lRx and
take c> O. Then an c-bracket for f relative to £ is a pair (f', f") E £i x £t
such that:
a) f' ~ f ~ I" ;
b) £i(f') , £f(f") E lR;
We put
_ { -xl For each c > 0 there is an } £(£):= f E lR
c -bracket for f relative to £ o
Proposition 2.3.8 Let (X, £, £) be a Daniell space. Then, given f E £(€) ,
SUp{£i(g) I 9 E £i, 9 ~ f} = inf{€t(g) I 9 E £t, 9 ~ f} E lR.
Proof. Let c > 0 be given. There is an c-bracket (f',I") for f relative to €.
Using Proposition 6, we have
-00 < €i(f') ~ sup{€i(g) I 9 E £i, 9 ~ f}
~ inf { £f (g) I 9 E £t , 9 ~ f}
~ £t(f") < 00.
But €t(f") - €i(f') ~ c and c > 0 is arbitrary, so
SUp{£i(g) I 9 E £i, 9 ~ f} = inf{€t(g) I 9 E £t, 9 ~ f} E lR. 0
2.3 The Closure of a Daniell Space 313
Definition 2.3.9 Given a Daniell space (X,.c, €) , put
f : C( €) ---+ IR ,
f t----+ sup{€.J.(g) I 9 E.c.J., 9 ~ J} = inf{et(g) I 9 E.ct , 9 ~ J}.
The triple (X,C(€),f) is called the closure of (X,.c,€). o
We justify the terminology chosen by showing that (X, C(€),'l) is the small
est closed Daniell space extending (X,.c, €) .
Theorem 2.3.10 Let (X,.c, €) be a Daniell space. Then (X, C(€), f) is a
closed Daniell space extending (X,.c, €) .
Proof. We first show that C(€) is a Riesz lattice and that f is a positive linear
functional on C(€).
Take f, 9 E C(€) , h E (J -i- g) and c > O. Let (J',1") and (g', gil) be
~~brackets for f and g, respectively. Then
J'(x) + g'(x) ~ h(x) ~ 1"(x) + g"(X) (1)
for every x for which f(x) + g(x) is defined. Suppose now that f(x) + g(x) is
not defined. Since f'(x) f:- 00, g'(x) f:- 00 and 1"(x) f:- -00, g"(X) f:- -00, we
conclude that
J'(x) + g'(x) = -00, J"(x) + g"(X) = 00,
and thus (1) holds in this case too. Moreover
and
Thus (J' + g', 1" + gil) is an c~bracket for h relative to €. It follows that
h E C( €) , and since
€.J.(J' + g') ~ f(J) + f(g) ~ €t(J" + gil) ,
it also follows that
314 2. Elementary Integration Theory
£(h) = £(1) + £(g) .
Similarly, one can prove that given a E ffi, al belongs to C(£) and £(aJ) =
a£(I) . Furthermore,
£t(l" V g") + £t(l" 1\ g") = £t(l" + g") E ffi
and therefore the individual terms on the left-hand sides must be real. Subtract
the first equation from the second one to obtain
(£t(l" V g") - £.1.(1' V g')) + (£t(l" 1\ g") - £.1.(1' 1\ g'))
Since both of the first bracketed expressions are positive (Proposition 6), we
conclude that (I' V g' , 1" V g") is an c-bracket for I V 9 , and (1'1\ g' , 1" 1\ g") is an c-bracket for 11\ g. Hence I V 9 E C(£) and 11\ 9 E C(£) .
The positivity of e follows from the relation £(1) 2': £-1-(0) = 0, which holds
for every IE C(£)+. To prove that (X, C(£), £) is a closed Daniell space, it suffices to show that
for any increasing e-sequence (In)nEIN from C(£)+, 1:= V In belongs to nEIN
C(£) and
£(1) = sup £(In) (2) nEIN
(Proposition 2.2.9 c) => a)). So take c > O. For n E IN, let (I~,I~) be an
(c/2n +1)-bracket for In, relative to £. Then ( V lIt, V if) is a ( E ~)-k<n k<n k<n
bracket for V Ik, that is, for In. This follows easily by complete i~duction, k<n
adapting the -arguments used above to show that (I' V g' , 1" V g") is an c-
bracket for I V g. In particular, for n E IN we have
Put
1":= V I:. nEIN
Using Proposition 2 f), we obtain
2.3 The Closure of a Daniell Space 315
Since the sequence (Z(Jn))nEIN is bounded in IR, £t(J") is real. The function
I" will serve as one of the two functions in the desired E-bracket for f. To
obtain the other function, first choose m in 1N with
(4)
Now choose l' from £.). so that l' :::; fm, £.).(J') is real and
(5)
The function f' will do. From (4) and (5) it follows that
(6)
Now (3) and (6) imply that
£.).(J'):::; sup£(Jn):::; £t(J") nEIN
and
Clearly l' :::; f :::; I" , so (J',I") is in fact an E-bracket for f, relative to £.
Since E was arbitrary, we conclude that f belongs to £(£) and (2) holds.
It remains to show that (X, £ (£), £) extends the original Daniell space
(X, £, £) . So take f E £. The set
A:= {If I = oo}
is £ exceptional and so e A belongs to £ n IR x . Hence e A belongs to both £t
and £.)., and £t(eA) = £.).(eA) = £(eA) = O. We conclude that eA belongs to
£(£) and that
Thus A is £(£)-exceptional (Corollary 2.2.12 b) => a)). We now use Propo
sitions 2.1.21 a),b), and 2.2.3 b). The function feA belongs to N(£(£)) and
therefore to £(£). The function feX\A belongs to £ n IRx and therefore to
£(£) . Moreover,
316 2. Elementary Integration Theory
Since f is the sum of feX\A and feA, f belongs to C(f) and
We have shown that £ C C(f) and flc = f, completing the proof. 0
Proposition 2.3.11 Let (X, £,f) be a Daniell space and (X, C',l') an ar
bitrary closed triple extending (X, £, f). Then the following implications hold.
a) If f E £t and tt(J) < 00, then f E £' and l'(J) = tt(J) .
b) If f E £i and fi(J) > -00, then f E £' and f'(J) = fi(J) .
Proof. In view of Proposition 4, it suffices to prove a). Take f E £t with
ft(J) < 00. There is an increasing sequence (In)nEIN from £ n lRx having f as its supremum. Then
sup ['(In) = sup f(Jn) = ft(J) < 00. nEIN nEIN
Hence (fn)nEIN is an increasing £I-sequence, and since (X, C',l') is closed, a)
follows. 0
The following theorem implies in particular that the closure of a closed
Daniell space is the given Daniell space itself.
Theorem 2.3.12 For every Daniell space (X, £,l) , (X, C(f),7!) is the small
est closed Daniell space extending (X, £,f) .
Proof. Let (X, C',l') be a closed Daniell space such that (X, £,f) ~ (X, C',l')
and take f E C(f) . There is an increasing sequence (f~)nEIN from £i and a
decreasing sequence (f~)nEIN from £t such that the sequences (fi(f~»nEIN and
(tt (f~»nEIN both lie in lR,
V f~ 5, f 5, 1\ f~ nEIN nEIN
and
sup fi(f~) = inf £f(J~) = 7!(J) . nEIN nEIN
Using Proposition 11, we conclude that the sequences (J~)nEIN and (J~)nEIN
are l'-sequences. If we put
2.3 The Closure of a Daniell Space 317
f':= V f~, 1":= 1\ f~ , nE~ nE~
then f' ::; f ::; 1" , and the functions 1',1" both belong to .12', since (X,.c', £') is closed. Moreover, by Proposition 11,
We leave the proofs of the next two propositions to the reader.
Proposition 2.3.17 Let (X, £, f) be a Daniell space. Then for every f E -x rn.,
Proposition 2.3.18 Let (X, £, f) be a closed Daniell space and take -x
f,g,h Ern..
a) If f E £, then f*(f) = f(f) .
b) If f E £J., then f*(f) = fJ.(f).
c) If f ::; g £-a.e., then f*(J) ~ f*(g) .
o
2.3 The Closure of a Daniell Space 321
d) If f = 9 .c-a.e., then £*(f) = £*(g).
e) If h(x) = f(x) + g(x) .c-a.e. and if £*(f) + £*(g) is defined, then
£*(h) ~ £*(f) + £*(g).
o
Closed Daniell spaces (X,.c, £) can be characterized in terms of the map
pings £* and £* .
Proposition 2.3.19 Let (X,.c, £) be a closed Daniell space.
a) .c = {f E IRx I £*(f) = £*(f) E IR}.
b) N(.c) = {f E IRx I £*(Ifl) = O}.
Proof. a) follows from Propositions 15 a) and 18 a) and the fact that (X,.c, £) is its own closure.
b) By Proposition 2.2.11,
N(.c) = {f E IRx I If I E .c and £(Ifl) = O} .
Moreover,
-x for every f E IR . Accordingly, b) follows from a). o
Proposition 2.3.20 For every Daniell space (X,.c, £) ,
Proof. In view of Proposition 17, it suffices to establish the first equality. That
£* ~ t follows immediately from the inclusion
which holds for every f E IRx. Given f E IRx , it follows that £*(f) = -* -* -* £ (f) whenever £ (f) = 00. Suppose that £ (f) is real. By Proposition 15 c), -* - -£ (f) = £(g) for some 9 E .c(£) with 9 ~ f. Then
£*(f) :::; £*(g) = £(g) = t(f).
322 2. Elementary Integration Theory
Finally, suppose that t (I) = -00. If 9 belongs to £(£)t and 9 2:: f, then -* -t £ (g) = £ (g) > -00. Then, using what has already been shown, we have
£*(1) ::; £*(g) = t(g) = i(g).
Taking the infimum over all such 9 yields £* (I) = -00. o
Combining Propositions 19 and 20, we obtain a new characterization of the
closure (X, £(£), £) of a Daniell space in terms of the mappings £* and £* .
Theorem 2.3.21 Let (X, £, f) be a Daniell space.
a) If f E £(£) 1 then £(1) = £*(1) = £*(1).
b) £(£) = U E lRx 1£*(1) = £*(1) E lR}.
c) N(£(£)) = U E lRx I £*(Ifl) = O}. o
We now describe several aspects of summability which will be useful later.
Example 2.3.22 Take
F(X) := U E lRx I U f= O} is finite},
p.)( : F(X) ---+ lR, f I--t L f(x). xE{NO}
(X, F(X), £)() is a Daniell space. We claim that (X, £l(X), L:) is its closure.
By Example 2.2.10, (X,£l(X), L:) is a closed Daniell space. It obviously
extends (X, F(X), £)() . Take f E £l(X)+. By Example 1.1.11 d), U f= O} is
countable. Let (An)nEIN be an increasing sequence of finite subsets of X with
U An = U f= O}. For n E IN, put fn := feA n • (In)nEiN is an increasing nEIN sequence from F(X)+ with supremum f. Hence f E F(X)t and
£t(l) = sup £)((In) = sup L f(x) = L f(x) < 00. nEIN nEIN xEAn xEX
Proposition 11 a) implies that f E £(£)(). It follows easily that P(X) c £(£)().
The claim is now a consequence of Theorem 12. 0
The summability naturally extends to infinite families of real numbers the
notion of sum. Recall that a family of real numbers and a real-valued function
are two descriptions of the same object (see the section on preliminaries).
2.3 The Closure of a Daniell Space 323
Definition 2.3.23 As before, we denote by 2:: the functional
C1 (X) -tIR, ff-'t 2)1):= Lf(x). xEX
A function f E IRx is said to be summable iff it belongs to Cl(X). -x
For f E lR ,we write
2:* f(x) := L* (J). xEX
Equivalently, a family (aX)XEX from lR is said to be summable iff it belongs
to Cl(X). For every summable family (ax)xEX from lR, we write
LaX := L((ax)xEX) XEX
and we call 2:: ax the sum of the family (aX)XEX. In a similar way, given a xEX
family (ax)xEX from lR, we write
and we call 2::* ax the upper sum of the family. xEX
o
Definition 2.3.24 A family (JJ'EI from lRx is said to be summable iff for
each x EX, the family (J,(X))'EI is summable. If (J,)'EI is a summablefamily -x
from lR ,then 2:: f" defined by 'EI
Lf,: X -t lR, x f-'t Lf,(x) ~l ~l
is called the sum of the family (j,),o. If (J,),o is an arbitrary family from -x * lR ,then 2:: f" defined by
'EI
L* f, : X -t lR, x f-'t 2:* f,(x) 'EI 'EI
is called the upper sum of the family (J')'EI. o
Corollary 2.3.25 If (ax)xEX is a summable family from lR, then
xEX xEX
Similarly, if (J')'EI is a summable family from lRx , then
2:* f, = Lf,. o 'EI 'EI
324 2. Elementary Integration Theory
Proposition 2.3.26 Take f E m.x . Then the following are equivalent.
a) ~* f(x) Em.. xEX
Proof Only a) ::::} b) requires a proof. Let a := ~* f(x) Em.. There is a xEX
decreasing sequence (gn)nEIN in £1(X)t such that gn ~ f and ~t(gn) is real
for every n E:IN and
By Proposition 11 a), (gn)nEIN is a ~-sequence in £1(X), and
Et (gn) = E gn(x) xEX
for every n E :IN. Putting g:= /\ gn, we conclude that 9 E £1 (X) and f ~ g. nEIN
We claim that f = g. Suppose that f(x) < g(x) for some x EX. Put
c5:= (g(x) - f(x)) /\ 1
and choose n E :IN such that
Then h:= gn - c5e{x} E £1(X) and h ~ f, but
contradicting the monotonicity of the mapping ~ . o
Corollary 2.3.27 For every family (ax)xEX from m.+,
Proof. If (aX)XEx is summable, the assertion follows from the definitions.
Otherwise it follows from Proposition 26. 0
The next proposition collects some special properties of summable se
quences.
2.3 The Closure of a Daniell Space 325
Proposition 2.3.28 Let (an)nEIN be a summable sequence of real numbers.
m
a) E an = lim E ak· nEIN m--+oo k= 1
b) inf (E lam I) = 0 . nEIN m>n
c) lim an = O. n--+oo
Proof. a) Apply the Lebesgue Dominated Convergence Theorem to the se
quence (In)nEIN from fl(IN) defined by
n
fn := L ake{k} (n E IN), k=l
using the fact that (lanl)nEIN is summable. b) follows from a) , and c) follows from b). o
Corollary 2.3.29 A sequence (an)nEIN of real numbers is summable iff
n
lim '" lakl < 00. n-t-oo L....J o
k=l
Corollary 2.3.30 For every sequence (an)nEIN from IR+,
o
We continue with some applications of the preceding considerations.
Theorem 2.3.31 Let (X, C, f) be a closed Daniell space and (J.}.El a count
able family from C. Then the following are equivalent.
a) The family (f(lf.I)'EI is summable.
b) ~lf.1 belongs to C . • EI
If these assertions hold, then so do the following.
c) f (~lf.l) = Ef(lf.l)· .EI .EI
d) The family (f.).EI is summable C-a. e.
e) The family (f(J'»'E/ is summable.
326 2. Elementary Integration Theory
f) For f E lRx , if f = L* f, .c-a.e., then f belongs to .c and £(J) = 'EI
~ £(J,). 'EI
Proof We may suppose that I is infinite. Let <p: IN -+ I be bijective. Put
A:= U{II,I = oo}. 'EI
Note that A is .c-exceptional.
a) ::::} b), c). The sequence ( ~ * If cp(k) I)nElN , whose terms belong to .c, is kElNn
increasing and
o :::; £ ( L* Ifcp(kl I) = t £(Ifcp(k) I) :::; L £(11,1) < 00.
kElN n k=! 'EI
Thus (~*lfcp(k)l) is an £-sequence from .c. Therefore Llfcp(k)1 be-kElN n nElN kElN
longs to .c and, using Corollary 30,
£ (Llfcp(kll) = L £(lfcp(k) I) = L £(If,I)· kElN kElN 'EI
Since L*lf,1 is equal to Llfcp(d, both b) and c) follow. 'EI kElN
b) ::::} a). For every n in IN, ~*Ifcp(d belongs to .c and kElN n
It follows that (£(lfcp(k)l)hElN is sum mabIe (Corollary 29). So, therefore, is
(£(11,1)),0,
b) ::::} d). Since the function L* If,1 belongs to .c, it is real .c-a.e. Thus 'EI
d) follows from Proposition 26 a) ::::} b).
a) ::::} e). Since I£(J,) I :::; £(If,1) for every ~, e) follows easily from a)
(Example 1.3.15).
a),b),d) ::::} f). In view of Proposition 2.2.3 d), it suffices to show that ~* f, 'EI
belongs to .c and
£ (2:: I,) = L £(J.) . 'EI 'EI
Set
2.3 The Closure of a Daniell Space 327
B := {x E X I {lft{x)J)tEI is not summable}
and note that B is C-exceptional. We have
For n E IN,
n
~ ft = lim '" f<p(k)eX\B C-a.e. L n-+ooL.-t tEl k=l
Itf<p(k)eX\B1 ~ Llftex\BI ~ L*lftl. k=l tEl tEl
Thus E* ft is the C-a.e. limit of a sequence from C bounded in C. Using the tEl
Dominated Convergence Theorem, we conclude that ~ ft belongs to C and tEl
Corollary 2.3.32 Let (X, C, f) be a closed Daniell space. Suppose, for some
sequence (fn)nEIN from C and for some function f belonging to C, that the
sequence (f(lf ~ fnl))nEIN is summable. Then the sequence (fn)nEIN is bounded
in C, and
f{x) = lim fn(x) C-a.e. n-->oo
Proof. Define
hn : X ----+ lR , .-----r - X' ----' { oof{X) - fn(x) if defined
otherwise,
(7)
Then (flhnJ))nEIN is summable. By Theorem 31 a) => b) & d), (hn)nEIN is
summable C-a.e. and h belongs to C. At each x for which (hn(X))nEIN is
summable we have
lim (fn{X) - f(x)) = 0 n-->oo
so (7) holds. Evidently,
-(h + If I) ~ fn ~ h + If I
for every n E IN. Since h + If I belongs to C, the sequence (fn)nEIN is bounded
in C. 0
328 2. Elementary Integration Theory
Corollary 2.3.33 Let (X, C, f) be a Daniell space. Then every function in
"l(f) is the "l(f)-a.e. limit of a sequence from C that is bounded in "l(f).
Proof. Let f E "l(f). For each n in N there is a function gn belonging to "l(f) net such that f ::; gn and
f(gn) - f(j) = ft(gn) - f(j) < 2 \n (Proposition 11 a)), and there is a function fn belonging to en IRX such that
Since the sequence (1/2n)nEIN is summable, the sequence mil - fn!})nEIN is summable. Now apply Corollary 32 to conclude that the sequence (jn)nEIN is
bounded in "l(f) and
f(x) = lim fn(x) "l(f)-a.e. n-+oo
Proposition 2.3.34 Let (X, C, f) be a closed Daniell space. Then
-x for every family (j')'EI from IR+ , and
-x for every countable family (j')'EI from IR+ .
o
(8)
(9)
Proof For (j')'EI a finite family from IR~ , inequalities (8) and (9) follow by
complete induction from Propositions 18 e) and 15 g).
Let (j,),E/ be a count ably infinite family from IR: and cp : N -+ I a bijection. Using (9) for finite families, together with Theorem 16 a) and Corollary
30, we have
2.3 The Closure of a Daniell Space 329
e (E'I.) = e (2:* Icp(n») = e (sup 2:* ICP(k») = .EI nEIN nEIN kEINn
Finally, let (J.).EI be an arbitrary family from IR: . According to Corollary 27,
E' 4 (f.) = sup L* 4 (f.) . • EI /fi~[te .EJ
For every finite subset J of I,
Inequality (8) follows. o
330 2. Elementary Integration Theory
2.4 The Integral for a Daniell Space
The following notation simplifies the statements of many later results.
Definition 2.4.1 Let Fe IRx . Then
9t(F) := {U =J O} I f E F} . o
Proposition 2.4.2 For every Riesz lattice C, the following hold.
a) U < g} E 9t(C) whenever f, 9 E C.
b) Given A E 9t(C) , there is an f E C+ n IRx with A = U > O}.
c) Au B E 9t(C) and An B E 9t(C) whenever A, B E 9t(C) .
d) If (X, c, £) is a Daniell space, then for every A E 9t(C(f)) there is a
sequence (An)nEIN in 9t(C) such that A c U An. nEIN
Now suppose that (X, C, £) is a closed Daniell space. Then
e) U An E &t(C) for every sequence (An)nEIN from 9t(C) . nEIN
f) If f E C and A E 9t(C) , then feA and feX\A belong to C.
g) A\B E 9t(C) whenever A, BE 9t(C) .
Proof. a) Note that U<g}={h/\O=JO} and hEC,whence h/\OEC,
where
_ { f(x) - g(x) h:X--tIR, Xl---t
o
if defined
otherwise.
b) There is agE C with A = {g =J O}. Put B := {Igl = co}. Then
eB E C and Iglex\B E C (Proposition 2.1.21 b)). Define f:= eB + Iglex\B.
c) Take f, 9 E C+ n IRx with A = U > O}, B = {g > O} and note that
Au B = U V 9 > A}, An B = {f /\ 9 > O}.
d) Given A E &t(C(£)), take f E C(£)+ with A = U > O}. There
is an increasing sequence (fn)nEIN in C+ n IRx such that f ~ V fn. Put nEIN
An := Un > O} for every n E IN .
2.4 The Integral for a Daniell Space 331
e) Let (An)nEIN be a sequence in ~ (.c) . Given n E IN, there is an fn E .c+
such that An = Un > O} . Multiplying by suitable numbers if necessary, we may
assume that eUn) < 1/2n. By Theorem 2.3.31 a) ::} b), f := L;* fn belongs nEIN
to .c and hence
U An = U # O} E ~ (.c) . nEIN
f) First suppose that f E .c+. Take 9 E.c+ with A = {g > O} . Then
feA = V U Ang) E.c. nEIN
Now if f E .c is arbitrary, then feA = f+eA - f-eA E .c. Moreover,
feX\A E (f ..:.. feA) C .c.
g) Take f E .c+ n IRx with A = U > O}. By f), feB E .c. Hence
A\B = U - feB # O} E 6i (.c). o
We now introduce the functions and sets which will serve as the exceptional
functions and sets for the integral.
Definition 2.4.3 For (X,.c, e) a Daniell space,
N(e):= U E IRx I feA E N(l(e)) for every A E ~(C(e))},
!Jl(e) := {B C X IBn A E !Jl(C(e)) for every A E ~ (C(e))}. 0
Proposition 2.4.4 Let (X,.c, f) be a Daniell space.
a) N(C(e)) c N(e) and !Jl(C(e)) c !Jl(e)).
b) Every subset of a set from !Jl(e) belongs itself to !Jl(e).
c) U An E !Jl(e) for every sequence (An)nEIN from !Jl(e). nEIN
-x - ~ d) N(e) = U E IR I feA E N(.c(e)) for every A E 9t(.c)}.
e) !Jl(e) = {B c X IBn A E !Jl(C(e)) for every A E ~(.c)}.
f) For every f E IRx , f E N(e) iff U # O} E !Jl(e) .
332 2. Elementary Integration Theory
Proof. We leave the proof to the reader. For d) and e), take Proposition 2 d)
into account. 0
Proposition 2.4.5 Let (X, £, £) be a Daniell space. Take g, hE £(£) with
{g =I h} E 1)1(£). Then £(g) = £(h) .
Proof. We have
{g =I h} c {g =I O} U {h =I O} E 6i (£(£)).
Consequently, {g =I h} E 1)1(£(£)). o
Definition 2.4.6 Let (X, £, £) be a Daniell space. Then
£1(£) := {J E IRX I {J =I g} E 1)1(£) lor some 9 E £(£)}.
For IE £1(£), we put
where 9 E £(£) , {J =I g} E 1)1(£). (1)
(In view 01 the preceding proposition, Id is well-defined.) Functions belonging
to £1(£) are said to be £-integrable. The number Itl is called the £-integral
01 I· The triple
(X, £1(£),1)
is called the integral lor the Daniell space (X, £, f) . o
The sign I was introduced by Leibniz in 1675. The name integral was used
for the first time by Jakob Bernoulli in 1690.
Note that given I E £1 (£) , the function 9 appearing in (1) can be chosen in
£(£) nIRx , because {Igl = Do} is £(£)-exceptional. Moreover, if IE £1(£)+,
then 9 can be chosen in £(£)+, since
{J V 0 =I 9 V O} C {J =I g} E 1)1(£)
and 1= IvO.
Theorem 2.4.7 For every Daniell space (X, £, £), (X, £1(£), It) is a closed
Daniell space extending (X, £, £) . Moreover,
2.4 The Integral for a Daniell Space 333
Proof. Let III 12 E .c1(1!) , and take 9I. g2 E £(I!) n JRx such that {II =F gd E
1J1(1!) and {l2 =F g2} E 1J1(1!) . Then
{II V 12 =F gl V g2} C {II =F gd u {h =F g2} E 1J1(1!) .
Consequently, II V 12 E .c1(1!). Similarly, 11/\12 E .c1(1!). Take hE (II -i- h)· Then
and thus hE .c1(1!). Moreover,
Ih = f(gl + g2) = f(gd + f(g2) = 111 + 112.
We leave it to the reader to verify that OIl E .c1(1!) and ftOlI = OIftl whenever
IE .cl (£) and 01 E JR, and that ftl ~ 0 whenever IE .c1(1!)+. Consequently, .c1(1!) is a Riesz lattice in JRx and Ii is a positive linear functional on .cl (£).
Let (fn)nEIN be an increasing Irsequence from .c1(1!)+. Given n E IN' ,
choose g~ E £(I!)+ such that {In =F g~} E 1J1(1!) . Put
gn:= V g;" m~n
for every n E IN' . Then
{In =F gn} C U {1m =F g;"} m$n
for every n E IN' , so {In =F gn} E '.n(I!) (Proposition 4 b),c)). Put
B:= U {In =F gn} . nEil"
By Proposition 4 c), B E '.n(I!). The sequence (gn)nEIN is an increasing f
sequence in £(I!) , and thus
g:= V gn E £(I!). nEil"
But
{V In =F g} c B E '.n(£) , nEil"
and so V In E .c1(1!). Moreover, nEil"
334 2. Elementary Integration Theory
1 (V fn) = £(g) = sup £(gn) = sup lfn-f nEIN nEIN nEIN f
In view of Proposition 2.2.9 c) =} a), (X, £1(£), fe) is a closed Daniell space.
It clearly extends (X, £(£),£) and hence also (X, £, e).
Let us now prove that SJ1(£I(£)) = SJ1(£). Take A E SJ1(£I(£)). Then feeA =
o and there is agE £(e)+ such that {g =f. eA} E SJ1(e). Since £(g) = 0, we
have 9 E N(£(e)) (Proposition 2.2.11 b) =} a)). Hence {g =f. O} E SJ1(£(£)), and since
it follows that A E SJ1( £) . Conversely, if A E SJ1( e) , then {ooe A =f. O} E SJ1( £) , and since 0 E £(e) , ooeA E £I(e). In other words, A E SJ1(£I(e)).
Finally, use Proposition 4 f) to prove the last statement of the theorem. 0
Definition 2.4.8 Functions belonging to N(e) , that is, to N(£l(e)) , are
called £-exceptional functions. Sets belonging to SJ1(e) , that is, to SJ1(£l(e)) , are called e --exceptional sets. In other words, we call the £1 (e) -exceptional
sets and functions simply £ -exceptional.
A property P that refers to elements of X is said to hold e-almost every
where (or simply, £-a.e.) iff it holds £I(£)-almost everywhere. Rather than
write P £l(e)-a.e. and P(x) £l(e)-a.e., we write, respectively, P e-a.e. and
P(x) e-a.e. 0
In view of Theorem 4.1.12 (b) of "Integration Theory, Vol. I" by C. Con
stantinescu and K. Weber (in collaboration wit A. Sontag), the definition of
integral given there (Definition 4.2.12) coincides with the present definition
(Definition 6). It was shown in that book that the integral defined there is, in
fact, the largest (not only a maximal!) element in a natural set of extensions
of the given Daniell space. We shall neither use this result nor repeat its proof
here. What we do prove is that applying the integral construction to the Daniell
space (X, £1 (e), Ie) , we do not obtain a further extension.
Theorem 2.4.9 For every Daniell space (X, £, £), the Daniell space
(X,£I(£),If) is its own integral.
Proof. It clearly suffices to show that £1(Je) is contained in £1(£). We prove
first that SJ1(Je) C SJ1(e). Take BE SJ1(Jf) and A E 9l (£(e)). Then, taking into
account that
2.4 The Integral for a Daniell Space 335
we have
and thus
B n A = (B n A) n A E 1J1(£(£)).
Since A is arbitrary, we conclude that B E 1J1(l') .
Now take f E £.1(ft). In view of (2), there is agE £1(£) such that
It follows that f = 9 £-a.e., and Proposition 2.1.21 c) implies that f E £1(l'). o
We provide a simple example for the integral construction.
Example 2.4.10 Consider the Daniell space (X, £, £) from Example 2.3.13.
The reader will readily verify the following statements.
N(£(£))
1J1(£(l'))
N(£)
-x {f E IR I f(a) = f(b) = f(c) = O} ,
{A c X I A c {d, e} } ,
{f E IRx I fex\{c} E NCC(l'))}
{f E IRx I f(a) = f(b) = O} ,
1J1(£) = {A c X I A c {c,d,e}} ,
£1(l') {f E IRx I f(a) = V(b) E IR},
ftf f(a) for every f E £1(£) .
Note that (X, £1(£), ft) is a proper extension of (X, £(£), 1). o
In numerous important cases, the integral and the closure of a Daniell space
are identical. We present a sufficient condition for this to be true.
Proposition 2.4.11 Let (X, £, £) be a Daniell space such that
336 2. Elementary Integration Theory
i) X = U Ai AE!){(.c)
ii) U B c U An for some sequence (An)nElN from 6i (.c) . BEIJI(C(£» nElN
Then
(X, £(f), f) = (X,.c 1 (f), [) .
Proof. It suffices to show that .c 1(f) c £(f), and this is an easy consequence
of the inclusion lJl( f) c lJl(£( f)) which we now prove. Take C E lJl( f). For
every A E 6i (.c) ,
An (C\ U An) E lJl(£(f)), nElN
and thus, in view of ii),
A n (C\ U An) = 0. nElN
Hence, by i), C\ U An = 0, that is, C c U An. Since C n An E lJl(£(f)) nElN nElN
for every n E IN, it follows that
C = U (C nAn) E lJl(£(f)). o nElN
Corollary 2.4.12 (X, f1 (X), L) is its own integral. o
Definition 2.4.13 Let:F c lRx . Then :F is said to be a-finite iff there is
a countable family (fL)LEI from :F such that
X =UUL :;fO}. o LEI
Corollary 2.4.14 Let (X,.c, f) be a Daniell space. If .c is a-finite, then
(X,£(f),f) = (X,.c 1(f), [) . o
The following properties of f-integrable functions will be useful later.
Proposition 2.4.15 Let (X,.c, f) be a Daniell space, and take f E .c 1(f) .
2.4 The Integral for a Daniell Space 337
a) There is an A E 6i (£(£)) such that
A c {f #- O}, feA E £(£) , fex\A E N(£) .
b) There are an increasing sequence (An)nEIN in 6i (.c) and a set C E 1)1(£)
such that
{f #- O} c (U An) U C . nEIN
c) feB E £(£) whenever B E 6t (£(£)) .
d) If {j #- O} c {g #- O} for some 9 E £(£) , then f E £(£) .
Proof. a) There is agE £(£) with {f #- g} E 1)1(£). Putting B := {g #- O},
we have
Since geB = 9 E £(£) , we conclude that feB E £(£). Moreover,
{jeX\B #- O} C {f #- g} E 1)1(£).
By Proposition 4 f), fex\B E N(£) . Now put A := {feB #- O} and note that
A C B and that
B\A c B n {f #- g} E 1)1(£(£)).
b) follows from a) and Proposition 2 d).
c) Choose A as in a). Then feB\A E N(£) and therefore
Moreover, in view of Proposition 2 f),
Accordingly,
feB = feB\A + feBnA E £(£).
d) follows from c) . o
338 2. Elementary Integration Theory
Proposition 2.4.16
Then Let (X,.c, f) be a Daniell space and take I E .c1 (f) .
i l = s~'p ileA + i!!.f ileA. t AE!Jt (.c) l AE!Jt (.c) I.
(3)
In particular, il I E .c1(f)+, then
(4)
Proof By Proposition 2 f), leA E .c1(f) whenever A E 9t(.c).
First suppose that IE .c1(f)+. Apply Proposition 15 b) to find an increas
ing sequence (An)nEIN in 9t (.c) such that
We have
and (4) follows.
I = V leAn f-a.e. nEIN
i l = sup ileAn::; sup ileA::; il l nEIN I. AE!Jt(C) l l
Now let IE .c1(f) be arbitrary. Put B:= {J > O}. Then
and hence An B E 9t (.c) whenever A E 9t (.c) . Thus
i r = s~ ireA ~ s~ ileA ~ s~ ileAnB = s~ il+eA. t AE!Jt (.c) t AE!Jt (.c) l AE!Jt (.c) t AE!Jt (C) t
It follows that
i l+ = sup ileA. l AE!Jt(.c) l
Accordingly,
and (3) follows. o
Note that Daniell spaces with identical closures have also the same inte
gral. In particular, the integrals for a Daniell space (X,.c, f) and its closure
2.4 The Integral for a Daniell Space 339
(X, £(£), £) coincide. As a consequence, (3) and (4) remain true if 6l (£) is
replaced by 6l (£( £)) .
The next result is often used when proving that every £-integrable function
has a certain property P. In fact, one can apply the theorem, putting
F : = {J E £ 1 (£) I f satisfies property P}.
Theorem 2.4.17 (Induction Principle) Let (X, £, £) be a Daniell space
and suppose that F is a subset of £1(£) satisfying the following conditions.
i) £ c F.
ii) If (fn)nEIN is an h-sequence from F, and if f E £1(£) satisfies
f(x) = lim fn(x) £-a.e., n-->oo
then f E F.
Proof. By i), and ii),
{J E £t I £t(f) < oo} c F,
{J E £~ I £~(f) > -oo} c F.
Take f E £(£). For each n E IN, there is a lin-bracket (f~, f~) for f
relative to £, and we may choose these brackets in such a way that (f~)nEIN is
increasing and (f~)nEIN is decreasing. By ii),
f' := V f~ E F, f":= 1\ f~ E F, nEIN nEIN
and since f = f' £--a.e. (Proposition 2.2.13), f E F. Hence £(£) C F. Now take f E £1(£). Then f is £-a.e. equal to some g E £(£). Use ii)
for the constant sequence (g)nEIN to conclude that f E F. It follows that
£1(£) c F. 0
340 2. Elementary Integration Theory
2.5 Systems of Sets, Step Functions, and Stone Lattices
We begin by studying the domains on which measures will be defined.
Definition 2.5.1 A set !Jt of sets is called a ring of sets or a set-ring iff
the following two conditions hold.
a) The empty set belongs to !Jt.
b) Both Au Band A\B belong to !Jt whenever A and B do.
A set-ring !Jt is called a c5 -ring iff
c) n An belongs to !Jt for every sequence (An)nEIN from !Jt nEIN
and it is called a a-ring iff
d) U An belongs to !Jt for every sequence (An)nEIN from !Jt. nEIN
For <5 an arbitrary set of sets, define
X(<5):= UA. AE6
If !Jt is a set-ring (c5 -ring, a-ring) and X is any set containing X (!Jt) ,
then we say that !Jt is a set-ring (c5-ring, a-ring) on X.
A set-ring on X is said to be a-finite iff X can be written as the union
of countably many elements of the set~ring. If!Jt is a a ~finite set~ring on X ,
we shall also say that the pair (X,!Jt) is a-finite. 0
Observe that if !Jt is a a-finite set-ring on X, then X = X (!Jt) .
Proposition 2.5.2 Let!Jt be a set-ring.
a) The union of every finite family from !Jt belongs to !Jt.
b) The intersection of every nonempty finite family from !Jt belongs to !Jt.
c) If A and B belong to !Jt, then so does their symmetric difference ALoB.
d) If!Jt is a c5-ring, then the intersection of every nonempty countable family
from !Jt belongs to !Jt.
e) If!Jt is a c5-ring and if (A')'EI is a countable family from !Jt such that
U A, c A for some A E !Jt, then U A, E !Jt . 'EI ,EI
2.5 Systems of Sets, Step Functions, and Stone Lattices 341
f) If ~ is a a-ring, then the union of every countable family from ~ belongs
to ~.
g) If ~ is a a-ring, then ~ is nlso a o-ring.
Proof a) This is proved by complete induction on the number of elements in
the family. Note that U A. = 0 and 0 is in ~. tE0
b) For arbitrary sets A and B,
An B = A\{A\B) .
Thus An B belongs to ~ whenever A and B do. The full claim now follows
by complete induction.
c),d),f). These are immediate from the definitions.
e) If 1= 0, there is nothing to prove. If not, use the identity
UAt = A\n{A\At). tEl tEl
g) Let (An)nEIN be a sequence from ~. Set
A:= U An nEIN
and observe that
n An=A\ U{A\An ).
nEIN nEIN
Using in turn the definitions of a-ring and set-ring, we conclude in succession
that A belongs to 9l, that each A\An belongs to ~, that U (A\An) belongs nEIN
to ~, and, finally, that n An belongs to 9l. nEIN
D
Example 2.5.3 The set consisting of all finite subsets of a fixed set X is
always a o-ring. It is a a-ring iff X is finite.
If X is an infinite set, then the set consisting of all subsets A of X for
which either A or X\A is finite is a ring of sets but not a o-ring. D
Every ring of sets has a useful decomposition property. Finite unions de
compose into disjoint finite unions, and the decomposition can be achieved one
set at a time.
342 2. Elementary Integration Theory
Proposition 2.5.4 Suppose that ~ is a ring of sets and that (A).EI is a
finite family from ~. Then there is in ~ a finite disjoint family (BI<)I<EK of
nonempty sets such that for each L in I there is a subset K. of K with
Proof. (Complete induction on the number n of elements of I) For n = 0 and n = 1, the claim is trivial. Assume that the claim is true
for some n E IN . Let (A.).EI be a family in ~ consisting of n + 1 elements.
Choose La E I and put J:= 1\ {La}. By assumption, there is a finite disjoint
family (CAhEL of nonempty elements of ~ such that for each L E J there is
a subset L. of L with
A.= U CA' AEL,
Then the nonempty elements of the set
{CA n Ao I A E L} U {CA \A. o I A E L} U {Ao \ U CAl AED
constitute the required family (BI<)I<EK.
The proofs of the next two results are easy and left to the reader.
D
Proposition 2.5.5 Let (~.).El be an arbitrary non empty family of set-rings
(of o-rings, or a-rings, respectively). Then
n~· .El
is also a set-ring (a o-ring, or a a-ring, respectively). D
The following corollary establishes the existence of a smallest set-ring (0-
ring, a-ring) containing a given set of sets.
Corollary 2.5.6 Let <5 be an arbitrary set of sets. Then there is exactly one
set-ring that contains <5 and is contained in every set-ring containing <5. The
set-ring in question is
n{~ c ~(X(<5)) I ~ is a set-ring containing 6}.
These assertions remain true if "set-ring" is replaced in all occurrences by "0-
ring" (or by "a-ring"). D
2.5 Systems of Sets, Step Functions, and Stone Lattices 343
Definition 2.5.7 Let 6 be a set of sets. The set-ring characterized in Corol
lary 6 is denoted by 6 r , and is called the set-ring generated by 6. The 8-
ring generated by 6 , denoted 66 , and the a-ring generated by 6, denoted
6"., are defined analogously. 0
Proposition 2.5.8 For every ring of sets v:t,
v:t6 c {A c X(v:t) I A c B for some BE v:t} .
Proof. Let
6 := {A C X(v:t) I A c B for some BE v:t}.
Then 6 is a 8-ring, and 6 contains v:t. Therefore v:t6 C 6 . o
Proposition 2.5.9 If v:t is a 8-ring, then v:tu is the set consisting of all
unions of increasing sequences from v:t.
Proof. Define
6 := {A I A = U An for some increasing sequence (An)nEIN from v:t}. nEIN
Certainly v:t c 6 c v:tu. If we show that 6 is a a-ring, then the proof
will be complete, since v:tu is contained in every a-ring that contains v:t.
Obviously 0 E 6. Let A and B belong to 6. There are in v:t increasing
sequences (An)nEIN and (Bn)nEIN such that A = U An and B = U Bn. nEIN nEIN
Now (An U Bn)nEIN is an increasing sequence from v:t, so its union, AU B, must belong to 6. Next observe that
A\B = (U An) \B = U (An\B). nEIN nEIN
The sequence (An \B)nEIN is increasing and its terms all belong to v:t, since
An \B = n (An \Bm) mEIN
and v:t is a 8-ring. It follows that A \B belongs to 6.
Finally, let (Cn)nEIN be a sequence from 6. For each n in IN, there is in
v:t an increasing sequence (Cn,m)mEIN whose union is Cn . For each m in IN,
put
344 2. Elementary Integration Theory
Dm:= U Cn,m' n:5m
The sequence (Dm)mEIN is increasing and its terms belong to Vl. Therefore its
union belongs to <5. But
so U Cn belongs to <5. Thus <5 is in fact a a-ring and !RO" = <5 . 0 nEIN
We continue by discussing some important properties of step functions.
Definition 2.5.10 A step function on X is a real function f, defined on
X, such that
(1) .EI
for some finite family (A.).EI of subsets of X and some family (a.).E/ of real
numbers.
Given a subset <5 of I.lJ(X) and a function f in rn.x , the function f is
called an <5 -step function on X iff it has a representation (1) for which every
A. belongs to <5. In this case, (l) is called an <5-representation of f. We
denote by .c~ the set of all <5-step functions on X.
Given a subset <5 of I.lJ(X) and an <5-step function f on X, if
(2)
and (B")"EK is a disjoint family of nonempty sets belonging to <5, then (2) is
called a disjoint <5-representation of the <5-step function f. 0
Proposition 2.5.11 Let!R be a set-ring on X and f an !R-step function
on X.
a) There is a disjoint Vl-representation for f.
b) If f ;::: 0 and if
(3) ,.EK
is an arbitrary disjoint Vl-representation of f , then /3,. ;::: 0 for every K
in K.
c) If f = 0 and if (3) is a disjoint Vl-representation of f, then /3,. = 0 for
every K in K.
2.5 Systems of Sets, Step Functions, and Stone Lattices 345
Proof. a) Let
be an 9'\-representation of f. By Proposition 4, there are a finite disjoint family
(B".)"EK of nonempty sets in 9'\ and an associated family (KL)LEl of subsets of
K such that
for every L in I. For", E K, put
(3 .. := L aL' .el
Then
is a disjoint !Jl-representation of f. b),c) are easy to see.
BKCA~
o
Corollary 2.5.12 Let 9'\ be a set-ring on X and f an 9'\-step function on
X.
a) {f > a} E 9'\ for every real number a ~ o.
b) {f ~ a} E 9'\ for every real number a > 0 .
Proof. a) Let
be a disjoint 9'\-representation for f. Then, if a ~ 0 ,
b) can be proved similarly. o
Theorem 2.5.13 Let 9'\ be a set-ring on X.
a) The set of all 9'\-step functions on X is a real Riesz lattice.
346 2. Elementary Integration Theory
b) If f is an ~-step function on X, then so is the function f /\ ex.
Proof a) It is obvious that .c~ is a real vector space. To prove that .c~ is a
Riesz lattice, it is therefore sufficient, in view of Theorem 1.2.6 n),o), to show
that If I E.c~ whenever f E .c~ . But if
(4)
is a disjoint representation for f, then
Ifl = L 1,BltleB. E .c~ . ItEK
b) Let (4) be a disjoint V't-representation of f. Then, in view of a),
f /\ ex = f /\ e U B. E .c~ . o .EK
Definition 2.5.14 For.c a Riesz lattice in IRx , we say that .c has the
Stone property iff f /\ ex belongs to .c for every f E .c. A Riesz lattice in
IR x that has the Stone property is also called a Stone lattice. A Daniell space
(X,.c, €) is said to have the Stone property iff its Riesz lattice .c has the Stone
property. 0
Thus for any set-ring V't on X, the set of ~-step functions on X forms
a Stone lattice. We shall demonstrate that to every Stone lattice .c whose
elements are step functions on X, there corresponds a set-ring V't(.c) c qJ(X)
such that .c is exactly the collection of all ~(.c)-step functions on X. The set
ring in question consists of those subsets of X whose characteristic functions
belong to .c.
-x Definition 2.5.15 For:F c 1R ,
V't(F) := {A C X leA E F} .
Proposition 2.5.16 Let .c be a Riesz lattice in IRx.
a) V't(.c) is a ring of sets.
o
-x b) If.c is conditionally a-completely embedded in IR , then ~(.c) is a
8-ring.
2.5 Systems of Sets, Step Functions, and Stone Lattices 347
c) If C is a-completely embedded in R X , then ~(C) is a a-ring.
Proof. a) The empty set belongs to ~(C) since C contains the zero function.
Suppose that A and B belong to ~(C). Then eA and eB both belong to C.
Since
it follows that Au Band A\B both belong to ~(C).
b) Let (An)nEIN be a sequence from ~(C). For n E 1N, set Bn := n Am. m:5n
Then /\ eBn belongs to C, by Proposition 2.1.12 a) =} c) . Since nEIN
e n An = /\ e Bn , nEIN nEIN
the set n An belongs to ~(C). nEIN
c) This proof is left to the reader. o
Theorem 2.5.17 If C is a Stone lattice whose elements are step functions
on X, then C is the set consisting of all ~(C) -step functions on X .
Proof. Since C is a Riesz lattice, it is obvious from the definition of ~(C)
that every ~(C)-step function on X belongs to C. We must show that every
element of C is an ~(C)-step function. It suffices to consider only positive
functions from C, since every real function f has the representation f+ - f- .
The zero function is trivially ~(C)-step, so suppose that f E C+ and f i= O.
Then
f(X)\{O} = {15k IkE INn}
for some n E IN and appropriate numbers 15k E R, where we may and do
assume that 15k < 15k+1 for every k in INn-I. Define
348 2. Elementary Integration Theory
Then
We prove inductively that each Ck belongs to !R(.c). Since
eCI = (~/) A ex
and .c has the Stone property, eCI belongs to .c and C1 is in !R(.c). Suppose
then that Ck belongs to !R(.c) for every k in lNm , m < n. Then
eCm +1 = (_I_g) A ex, 'Ym+l
n m
g:= L 'YkeCk = I - L 'YkeCk . k=m+l k=1
m
The inductive hypothesis ensures that L 'YkeCk belongs to .c. We conclude k=1
successively that g belongs to .c, that eCm +1 belongs to .c and that Cm +!
belongs to !R(.c). Thus C k belongs to !R(.c) for every k in lNn . 0
We now present some general properties of Stone lattices.
Proposition 2.5.18 Let (X,.c, i) be a Daniell space such that .c is a Stone
lattice. Then so are C(i) and .c1(i).
Proof We use the Induction Principle 2.4.17 to prove the second assertion. Let
According to the hypothesis, .c c :F. Let (fn)nEJN be an Jrsequence from :F -x
and I E IR such that
I = lim In i-a.e. n-+oo
If (fn)nEJN is increasing then Ik A ex :::; lim In for every k E IN. If (fn)nEIN is n-+oo
decreasing, then IkAeX ~ (lim In) AO for every k. In either case, (fnAex )nEJN n-+oo
is an It-sequence from .c1 (i) , so
( lim In) A ex = lim (fn A ex) E .c1(i). n--too n--too
Since
I A ex = (lim In) A ex i-a.e., n-+oo
2.5 Systems of Sets, Step Functions, and Stone Lattices 349
f 1\ ex E £1(£), and hence f E F. By Theorem 2.4.17, F = £1(£), showing
that £ 1 (£) has the Stone property.
The fact that C(£) is a Stone lattice, follows from the result just proven,
taking Proposition 2.4.15 d) into account. 0
Our next proposition describes the approximation of (arbitrary) positive
functions by positive step functions. We strongly recommend that the reader
sketch the first three or four members of the sequence (fn)nEJN mentioned in
this proposition.
Proposition 2.5.19 -x
Take f E 1R+ . For n in IN, define
Then (fn)nEJN is an increasing sequence of positive step functions on X whose
supremum is f.
Proof. Take x EX. If f(x) = 00, then fn(x) = n for every n E IN. It follows
that (fn(X))nEJN is increasing and has supremum 00.
Now suppose that f(x) < 00. Take n > f(x). There is a unique k E IN
with k/2n ~ f(x) < (k + 1)/2n. Then fn(x) = k/2n and hence 0 ~ f(x) -
fn(x) < 1/2n. It follows that f(x) = lim fn(x). To show that (fn(X))nEJN is n-+oo
an increasing sequence, take n E IN. If f(x) ~ k/2n for some k with 1 ~ k ~
n2n , then f(x) ~ 2k/2n+1 and 1 ~ 2k ~ n2n+l. Thus there are at least twice
as many j's for which f(x) ~ j /2n+1 and 1 ~ j ~ n2n+1 as there are k's for
which f(x) ~ k/2n and 1 ~ k ~ n2n . It follows that
-x Proposition 2.5.20 Let £ be a Stone lattice in 1R . Then f 1\ aex E £ whenever f E £ and a E 1R, a > 0 .
Proof. Simply note that
f 1\ aex = a ( (~f) 1\ ex) o
350 2. Elementary Integration Theory
Proposition 2.5.21 Let £ be a Stone lattice, conditionally a-completely em
bedded in IR x .
a) For every 1 E £ and every real number a > 0, the set U ~ a} belongs
to 9\(£).
b) Every positive function belonging to £ is the supremum of an increasing
sequence of positive 9\(£) -step functions on X .
c) Every set belonging to !R(£) can be written as the union of an increasing
sequence from !R(£).
Proof. a) follows from Proposition 20 and the identity
e{f>o} = 1\ n + 1 ((f /\ aex) - (1 /\ ~ex)) - nEIN a n + 1
b) In view of a), b) follows from Proposition 19.
c) Let A E !R(£), and choose 1 E £+ such that A = U i- O}. By b),
there is an increasing sequence (fn)nEIN of positive !R(£)-step functions on X
such that 1 = V In· Set An := Un > O} for every n E IN. Note that each nEIN
An belongs to !R(£) (Corollary 12 a)). Evidently, (An)nEIN is an increasing
sequence whose union is A. o
Corollary 2.5.22 Let (X, £1, l'.) and (X, £2, (2) be closed Daniell spaces
such that £1 and £2 are Stone lattices. Then the following are equivalent.
Proof. The implication a) => b) is trivial.
b) => a). Take f E (£1)+. Then 1 is the supremum of an increasing
sequence (fn)nEIN of positive !R(£I)-step functions on X (Proposition 21 b)).
This same sequence is both an (2-sequence from £2 and an (I-sequence from
£1 . We conclude that f belongs to £2 and
An arbitrary f in £1 can be written as 1+ - f- . Thus a) follows. 0
2.5 Systems of Sets, Step Functions, and Stone Lattices 351
Exercises
E 2.5.1 (E)
A monotone set is a set of sets, 6, with the property that if (An)nEIN is a
monotone sequence from 6, then lim An E 6 . n ..... oo
Prove the following.
(a) Let 6 be a set of sets. Then there is a monotone set 6 m such that
6 C 6 m and 6 m C 9'l for every monotone set 9'l with 6 C 9'l. 6 m is
uniquely determined by 6. 6 m is called the monotone set generated by
6.
(f3) Every a-ring is a monotone set.
(-y) If 6 is both a ring of sets and a monotone set, then 6 is a a-ring.
(6) Given the ring of sets, 9'l, 9'lm = 9'l" .
We provide some suggestions for (6) . Clearly 9'lm C 9'l" . Given A C X(9't) ,
define
6 A := {B C X(9'l) I A\B E 9'lm, B\A E 9'lm and Au BE 9'lm}.
Then, given A, B C X(9't), A E 6 8 iff B E 6 A . Given a monotone se
quence (An)nEIN from 6 A, lim An E 6 A . Hence 6 A is a monotone set. For n-+oo
A E 9't, 9'l C 6 A, and so 9'lm C 6 A . Consequently, if A E 9'l and B E 9'lm , then B E 6 A , and so A E 6 8 . As a result 9'lm C 6 8 whenever B E 9'lm, showing that 9'lm is a ring of sets. Using (-y), 9'l" c 9'lm.
A conditionally monotone set is a set of sets, 6, with the property that
n An E 6 whenever (An)nEIN is a decreasing sequence from 6 and U An E nEIN nEIN
6 whenever (An)nEIN is an increasing bounded sequence from 6. By analogy to (a)-(8) , the following statements hold.
(0:') Let 6 be a set of sets. Then there is a conditionally monotone set 6 mb
such that 6 C 6 mb and 6 mb C 9't for every conditionally monotone set
9'l with 6 C 9't. 6 mb is uniquely determined by 6. 6 mb is called the
conditionally monotone set generated by 6.
(f3') Every 6-ring is a conditionally monotone set.
("(') If 6 is a ring of sets and a conditionally monotone set, then 6 is a
6-ring.
(6') Given a ring of sets, 9't, 9'lmb = 9'l6 .
352 2. Elementary Integration Theory
2.6 Positive Measures
We begin with the definitions and elementary properties of various kinds of set
mappings, in particular, positive measures.
Definition 2.6.1 Let (5 be a set of sets. A mapping I-" : (5 -+ 1R is said to
be:
a) additive, if given A, B E (5 with Au BE (5 and An B = 0,
I-"(A u B) = I-"(A) + I-"(B) ;
b) positive, if I-"(A) ;::: 0 for every A E (5 ;
c) increasing, if it is increasing as a mapping from the ordered set (5 to the
ordered set 1R;
d) nullcontinuous, if
for every decreasing sequence (An)nEIN from (5 with
If !1t is a ring of sets and I-" : !1t -+ 1R is an additive, positive, nullcontinu
ous mapping, then I-" is called a positive measure on !1t. A positive measure
space (a positive 8- or a-measure space) is a triple (X, 9t, 1-") where X is
a set, !1t is a set-ring (a 8-ring or a a-ring, respectively) on X, and I-" is a
positive measure on !1t. o
Proposition 2.6.2 Let I-" : !1t -+ 1R be an additive mapping on the ring of
sets !1t. Then:
a) 1-"(0) = o.
b) If (A.).El is a finite disjoint family from !1t, then
c) Given A, B E!1t with A c B,
I-"(B\A) = I-"(B) - I-"(A) .
2.6 Positive Measures 353
d) J1- is positive iff it is increasing.
e) For all A, B E 9\,
and if J1- is positive, then
Proof. a) follows from the identity
b) can easily be proved by complete induction on the number of elements
in I. Observe that U A. = 0, so that b) follows from a) in case 1= 0. tE0
c) Note that B is the disjoint union of A and B\A. Since J1- is additive,
and c) follows.
d) If J1- is positive and A c B, then
and the inequality
follows from c) . Conversely, if J1- is increasing, then it follows from a) that
for every A in 9\.
e) Since Au B is the disjoint union of A and B\A, we have
J1-(A U B) = J1-(A) + J1-(B\A) .
Writing B as the disjoint union of B n A and B\A, we have
and e) follows. o
354 2. Elementary Integration Theory
Proposition 2.6.3 Let p, : 9't --t 1R be a positive, additive mapping on the
ring of sets 9't. Then the following are equivalent.
a) p, is nullcontinuous.
b) inf P,(An) = 0 for every decreasing sequence (An)nEIN from 9't with nEIN
n An = 0. nEIN
c) For every increasing sequence (An)nEIN from 9't, if U An belongs to 9't, nEIN
then
d) For every decreasing sequence (Bn)nEIN from 9't, if n Bn belongs to nEIN
9't, then
Proof. a) - b) follows from the fact that p, is increasing.
b) ~ c). Mimic the proof of Proposition 2.2.7 b) ~ c).
c) ~ d). Let (Bn)nEIN be a decreasing sequence from 9't and assume that
B:= n Bn nEIN
belongs to 9't. The sequence (B1 \Bn)nEIN is increasing, lies in 9't and its union
belongs to 9't:
nEIN nEIN
From c) we conclude that
Hence
from which it follows that
2.6 Positive Measures 355
d) ::::} b) is trivial. o
For positive set mappings, nullcontinuity and finite additivity together are
equivalent to countable additivity:
Theorem 2.6.4 Let p, : !Jt -+ IR be a positive mapping on the ring of sets
!Jt. Then the following are equivalent.
a) p, is a positive measure on !Jt.
b) Countable additivity. For every countable disjoint family (A,),El from
!Jt, if U A, belongs to !Jt, then the family (p,(A,) )'ET is summable and ,EI
c) For every disjoint sequence (An)nEIN from !Jt, if U An belongs to !Jt, nEIN
then the sequence (P,(An))nEIN is summable and
Each of these assertions implies that for any countable family (A,)'ET from !Jt,
whenever both sides are defined.
Proof. a)::::} b). Let (A')'EI be a countable disjoint family from !Jt whose
union belongs to !Jt. It suffices to consider the case where I is count ably infinite.
So let cp : IN -+ I be bijective. Construct an increasing sequence from !Jt whose
union is the same as U A, by putting Bn:= U A<p(k) for every n E IN . Since ,ET k<n
p, is nullcontinuous, additive, and positive, we have
00 > p, (UA,) = p, (U Bn) = supP,(Bn) = LET nEIN nEIN
356 2. Elementary Integration Theory
It follows that the sequence (Jl(Acp(k») kEIN is summable (Corollary 2.3.29) and
hence that the family (Jl(A')tEI is summable. Moreover
b) => c). This implication is evident.
c) => a). First notice that Jl(0) = o. Indeed, hypothesis c) together with
Proposition 2.3.28 a) implies that Jl(0) = lim nJl(0) , which is only possible if n .... oo
Jl(0) = o. The additivity of Jl then follows by hypothesis. We must verify that
Jl is nullcontinuous, for which we use the characterization in Proposition 3 c)
=> a). Let (An)nEIN be an increasing sequence from 9t whose union belongs
to !J't. Construct a disjoint sequence (Bn)nEIN in !J't with the same union as
the given sequence by putting Bl := Al and Bn := An \An- l for n E IN\{l}. Then
Jl (U An) = L Jl(Bn) = J~~ t Jl(Bk) = sup t Jl(Bk) = sup Jl(An) nEIN nEIN k=l nEIN k=l nEIN
as required. The reader can readily verify the final claim, observing that given a sequence
(An)nEIN in 9t,
is a disjoint sequence in 9t having the same union as (An)nEIN. o
Countable additivity appeared for the first time in the thesis of E. Borel
(1894). Next we turn to the notion of boundedness.
Definition 2.6.5 Let 9t be a ring of sets and Jl : 9t -t lR a positive additive
mapping. Then Jl is said to be
a) bounded if there is a real number a such that Jl(A) $ a for every A in
9t;
b) a-bounded if there is a countable family (A.)tEl from 9t with
inf Jl(A\A.) = 0 tEl
for every A in 9t.
2.6 Positive Measures 357
A positive measure space (X, 9l, Il) is called bounded or a -bounded iff the
measure Il is bounded or a-bounded, respectively. 0
Our first observation is an obvious consequence of the definitions.
Proposition 2.6.6 Every positive measure space (X, 9l, Il) for which the pair
(X,9l) is a-finite, is a-bounded. 0
We want to show that bounded positive measures are always a-bounded
and that positive measures on a-rings are always bounded. The proofs make
no use of nullcontinuity, so we formulate somewhat more general results first.
Proposition 2.6.7 Let 9l be a ring of sets, and Il : 9l -+ 1R a positive,
additive mapping.
a) If Il is bounded, then Il is a-bounded.
b) If 9l is a a-ring, then Il assumes a maximum on 9l; i.e., there is a set
B E 9l such that
sup Il(A) = Il(B) . (1) AE!)l
c) If 9l is a a-ring and B is any element of 9l for which (1) holds, then
Il(A \B) = 0 for every A E 9l.
Proof. a) Put
a := sup Il(A) AE!)l
and let (An)nEIN be a sequence from the set-ring 9l such that
Take A E 9l. Then A\An = (AUAn)\An and hence
for every n. Thus
Equality must hold throughout, so
358 2. Elementary Integration Theory
b) With Q and (An)nEIN as in the proof of a), put
Since BE 9t,
B:= U An. nEIN
Q = sup J.L(An) S; J.L(B) S; Q.
nEIN
It follows that Q = J.L(B) .
c) Using Proposition 2 c) and with Q as in the proof of a), we have
Q = J.L(B) S; J.L(B) + J.L(A\B) = J.L(A U B) S; Q
so J.L(A\B) = O. o
Corollary 2.6.8 Every bounded positive measure is a-bounded. Every positive
measure on a a-ring is bounded. 0
Example 2.6.9 Let 9t be a ring of sets and take x E X(9t). Define
Then IS~ is a bounded positive measure on 9t, called the Dirac measure
(1926) on 9t concentrated in x. 0
Example 2.6.10 Let;J' be the set of all finite subsets of X. Define
X:;J'--tlR, At---t LeA(x). xEA
Then (X,;J', X) is a positive IS-measure space and for A E ;J', X(A) is the
number of elements of A. For this reason, X is called the counting measure
on X. The measure X is bounded iff X is finite, and X is a-bounded iff X
is countable. o
Given a positive measure space (X, 9t, J.L) , we want to define a functional
e,.. on .c~ which is naturally associated with the positive measure J.L, where by
"naturally associated" we mean that
for every A in 9t. If
2.6 Positive Measures 359
is an 9\-step function on X , linearity requires that
f}J(J) = L Q.JL(A.) . (2) .EI
But if (2) is to be used as a definition for f}J' we need to know that the value of f}J (J) does not depend on the particular representation used for f.
Proposition 2.6.11 Let (X, 9\, JL) be a positive measure space. Suppose that
(A.).E1 and (BK.)K.EK are finite families from 9\ and that (Q.).EI and ((3K.)K.EK are families from 1R such that
L Q.eA, = L (3K.eB •. • EI K.EK
Then
Proof. First suppose that
We show that
By Proposition 2.5.4, there are a finite disjoint family (CAhEL of nonempty
sets in 9\ and an associated family (L.).EI of subsets of L such that for every
LEI,
AEL, AEL,
Take A ELand choose a fixed x E CA • Then
Thus
360 2. Elementary Integration Theory
.EI .EI AEL,
The general case follows from our special case by considering the 9t-step
function
2: a.eA, - 2: /3"eB •. o .EI "EK
Theorem 2.6.12 Given a positive measure space (X, 9t, J-t), there is exactly
one functional C; on £~ for which (X, £~, C;) is a Daniell space and
(3)
for every A E 9t. This functional is given by
.EI .EI
Proof By Proposition 11, C; (J) does not depend on the representation of f.
The functional C; is evidently linear and satisfies (3). Moreover, the positivity
of C; follows from that of J-t, taking Proposition 2.5.11 a),b) into account. The
uniqueness of the functional is trivial.
It remains to be verified that C; is nullcontinuous. Let (In)nEIN be a de
creasing sequence from £~ with
/\ fn = O. nEIN
Take € > O. For each n E IN, put An := {fn > €} and A := {II > O}. By
Corollary 2.5.12 a), A and all the An's are in 9t. Since (An)nEIN is decreasing
and n An = 0 , we conclude that nEIN
Put
Then
a:=supfl(x). xEX
2.6 Positive Measures 361
for every n E IN . Choose m E IN such that JL(Am) < c. Then
for each n ~ m. Since c is arbitrary,
o
Definition 2.6.13 For (X, 9l, JL) a positive measure space, the functional e;
and the Daniell space (X, £~, e;) described in Theorem 12 are called the func
tional associated with the measure JL and the Daniell space assocciated with
the measure space (X,!>l, JL) , respectively. 0
The following easily verified proposition describes the reverse situation.
Proposition 2.6.14 Let (X, £, £) be a Daniell space, and define
Then (X, !>l(£) , JLt ) is a positive measure space. o
-x Definition 2.6.15 For:F c IR. and e : :F ~ IR., the mapping
is called the set mapping induced (on 9l(:F)) bye. If (X, £, f) is a Daniell
space and JL is the set mapping induced on 9l(£) bye, then we call (X, 9l(£) , JL) the positive measure space induced by (X, £, e) and JL the positive meaBure
induced bye. 0
The Daniell spaces associated with positive measure spaces always have the
Stone property and their Riesz lattices always consist of step functions. Thus
there is no hope of recovering a Daniell space from its induced positive mea
sure space unless these two conditions hold. No other conditions are required,
however, as we see from the following theorem.
Theorem 2.6.16 Suppose that (X, £, e) is a Daniell space and £ is a Stone
lattice whose elements are step functions. Then there is a uniquely determined
positive measure space whose associated Daniell space is (X, £, e), namely the
positive measure space induced by (X, £, £) .
362 2. Elementary Integration Theory
Proof Denote by ,l the set mapping induced on !R(.c) by I. According to
Theorem 2.5.17, !R(.c) is the uniquely determined set-ring !R for which .c
is the set of !R-step functions. By Proposition 14, (X, !R(.c) , Ji) is a positive
measure space. The definition ,i(A) = l(eA) shows that (X, !R(.c) , ,i) is a positive measure space with which the Daniell space (X,.c, I) is associated. 0
Example 2.6.17 Consider counting measure X on the set X (Example 10).
Then f E 1Rx is an ~-step function iff {f "I O} is finite. For such a function
f, we have
Ix(f) = Ix ( L f(x)e{x}) = L f(x)lx(e{x}) = XE{f#O} XE{!#O}
= L f(x)x({x})= L f(x)=Lf(x). xE{!#O} XE{f#O} xEX
Note that we already know the closure of the Daniell space (X, .c~, Ix) as
sociated with (X,~, X): We showed in Example 2.3.22 that the closure is
(X, e1 (X), E). 0
Definition 2.6.18 The positive measure space (X,!R, p,) is called closed iff
i) U An E!R for every increasing sequence (An)nEIN from !R with nEIN
sup P,(An) < 00; nEIN
ii) I.l3(A) C !R for every A E !R with p,(A) = O. o
Proposition 2.6.19 If (X,!R, p,) is a closed positive measure space, then !R is a 6-ring.
Proof It suffices to show that the intersection of every decreasing sequence
from !R belongs to !R. This follows easily from the formula
o
Theorem 2.6.20 Let (X,!R, /-l) be a positive measure space. Then the follow
ing are equivalent.
a) (X,!R, p,) is closed.
2.6 Positive Measures 363
b) (X, v:t, Jl) is induced by a closed Daniell space (X,.c, f) for which .c is a
Stone lattice.
Moreover, if these equivalent conditions are satisfied, then there is only one
closed Daniell space with the Stone property which induces (X, v:t, Jl), namely
the closure (X, C(f;), f:) of the Daniell space (X, .c~, f;) associated with
(X, v:t, Jl) .
Proof. a)::::} b). Denote by (X, l, f) the closure of (X, .c~, f;) . By Theorem
2.5.13 b) and Proposition 2.5.18, l is a Stone lattice. We show that (X, l, i) induces (X, v:t, Jl) , that is, we must show that
v:t = v:t(l) (4)
and
(A E v:t) . (5)
The inclusion C in (4) and the equality (5) are trivial. To prove the reverse
inclusion in (4), take A E !Jt(l). There is a decreasing sequence (fn)nEIN in
(.c~)t such that fn ~ eA and (f;)t(fn) < 00 for every n E IN and such that
Given n E IN, fn = V hk for some increasing sequence (hk)kEIN from (.c~)+. kEIN
Put
(k E IN , m E IN\ { 1 } ) .
Then B km E v:t and for every m ~ 2 ,
Hence U Bkm E !Jt and, in view of Proposition 19, kEIN
An := Un ~ 1} = n U Bkm E!Jt.
By Proposition 19 again,
m;:::2 kEIN
B:= n An E v:t. nEIN
364 2. Elementary Integration Theory
Now A C Band
B\A C { A fn =I eA} E ')1(£). nEIN
If A E ')1(£) , then
I-'(B) = f(eB) = f(eA) = O.
Since (X, 9t, 1-') is closed, it follows that A E 9t. If A E 9t(£) is arbitrary,
then by the preceding argument B\A E 9t. Thus
A = B\(B\A) E 9t,
proving (4).
b) ::::} a) is a consequence of the definition of closed Daniell space and of
Corollary 2.2.12. For the final claim, let (X, C, £) be a closed Daniell space with the Stone
property which induces (X, 9t, 1-'). Then
9t(C) = 9t = 9t(£)
and
for every A E 9t. Corollary 2.5.22 b) ::::} a) implies that
(X, C, £) = (X, £, l) . o
Corollary 2.6.21 Let (X, 9t, 1-') be a closed measure space.
a) For every nonempty countable family (A.).EI from 9t which is directed
up, the conditions
sup I-'(A.) < 00 .EI
and
are equivalent and imply that
I-' (u A.) = sup I-'(A.) . • EI .EI
2.6 Positive Measures 365
b) For every nonempty countable family (A.).EI from !:R which is directed
down, the set n A. belongs to !:R and .EI
c) For every countable disjoint family (A).EI from !:R, the conditions
and
are equivalent and imply that
d) If (An)nEIN is a sequence from !:R that is bounded above in !:R, then the
sets
'.p(X) lim sup An and '.p(X) lim inf An n-+oo n-+oo
both belong to !:R. If, in addition, there is a set A satisfying
A = '.p(X) lim An, n-+oo
then A belongs to !:R and
Proof Each assertion can be verified by using the map A t-t eA to translate
the assertion in question into a familiar fact about closed Daniell spaces. In par
ticular, a) and b) follow from Theorem 2.2.17. c) is a consequence of Theorems 4 b) and 2.3.31 a) :::}b). Finally, d) follows from Corollary 2.2.20 and Theorem
2.2.21. 0
366 2. Elementary Integration Theory
Exercises
E 2.6.1 (E)
A semi-ring is a non empty set of sets, 9t, such that for A, B E 9t there are
finite disjoint families (A.),El and (B),);"EL from 9t with An B = U A, and 'El
A\B= U B).. ).EL
Prove that
(a) given the semi-ring, 9t, A E !)l,. iff there is a finite disjoint family (A')'EI
from 9t with A = U A, . 'El
(Let <5 denote the set of all unions of finite disjoint families from 9t.
Given A, B E <5 , first show that A n B E <5 , next prove that A \B E <5 and finally show that A U B E <5.)
We next describe an important example of a semi-ring. Let A i= 0 be an
interval of IR containing neither 00 nor -00. Let a be the left endpoint of A
and b its right endpoint. Note that -00 ::; a < b ::; 00. Given x, yEA with
x::; y, put
and let
{ [x,y[
[x,y 1:= [x,b]
if y i= b
if y = b,
J:={[x,yl Ix,yEA, x::;y}.
(J3) J is a semi-ring. The elements of Jr (i.e. the finite unions of intervals in
J) are called the interval forms on A.
Let 9t be a semi-ring. By a content on 9t we mean a function, J.L : 9t --+ IR , such that if A E 9t and if (A')'EI is a finite disjoint family from 9t with
A = U A, , then J.L(A) = ~ J.L(A,). J.L is called a-additive if given any disjoint 'El tEl
sequence (An)nEIN from 9t with U An E 9t, J.L( U An) = ~ /l(An). nEIN nEIN nEIN
Prove the following.
(-y) Given a content, J.L, on 9t, there is exactly one additive function ii on
!)l,. with iil!Jt = J.L .
(6) If J.L is a positive content, then ii is positive.
(c:) If J.L is a positive a-additive content, then ii is a positive measure.
2.6 Positive Measures 367
(() Given two additive functions on ~,J.l and II, such that J.l1!>t = 1I1!>t, then J.l = II.
(17) If 0 E ~ then each a-additive real function on ~ is a content.
For the proof of (c), we wish to show that jj is a-additive. First take
A E ~ and let (An)nEIN be a disjoint sequence in ~ with A = U An. nEIN
There is a finite disjoint family (B')'EI from ~ such that A = U B,. For ,EI
each LEI then, B, = U (An n B,). Thus for each n E IN and LEI, there nEIN
is a finite disjoint family from ~ whose union is An n B,. Bearing this in
mind, there are, for each LEI, a disjoint sequence (D,k)kEIN from ~ and a kn+l-l
sequence (kn)nEIN from IN such that kl = 1 and AnnB, = U D,k, so that k=kn
B, = U D,k. Use this to conclude that kEIN
J.l(B,) = E jj(An n B.), and finally, nEIN
that jj(A) = E J.l(An). nEIN
The general case is now an easy consequence of what has been proved.
E 2.6.2 (C)
Ulam Sets (S. Ulam, 1930) Let ~ be a ring of sets. A positive measure J.l on
~ is called a two-valued positive measure on ~ iff J.l(~) = {O, a}, a i- O. A set X is called a weak Ulam set (an Ulam set) iff there is no positive two-valued
measure J.l (no strictly positive measure J.l) on I.13(X) such that J.l( {x}) = 0 for every x EX.
Prove the following.
(a) For every set X, the following are equivalent.
(ad X is a weak Ulam set.
(a2) There is no nontrivial 8-stable ultrafilter on X.
(If ~ is a nontrivial 8-stable ultrafilter on X, consider the map
J.l: I.13(X) ---+ 1R, A t--+ { ~ if A E ~
if A ~~.
If J.l is a two-valued positive measure on I.13(X) , consider
~ := {A c X I J.l(A) i- O} .)
({3) If X is a weak Ulam set, then every set with the cardinality of X is also
a weak Ulam set.
368 2. Elementary Integration Theory
(-y) Every countable set is a weak Ulam set.
(8) Every subset of a weak Ulam set is a weak Ulam set.
(f) If I is a weak Ulam set and if (X,),El is a family of weak Ulam sets, then
U X, is a weak Ulam set. ,E/
(Put Z := {( L, x) I LEI, x E X.}. U X, has the same cardinality as some ,EI
subset of Z. Thus it is sufficient to show that Z is a weak Ulam set and then
to apply (13) and (8). So let p, be a two-valued measure on I.P(Z) with the
property that p,( {z}) = 0 whenever z E Z. For LEI, put Z, := {(L, x) I x E
X,} . Thus, for each L, Z, has the same cardinality as X, and is, consequently,
a weak Ulam set. Thus p,lz, = 0 for LEI. Given J c I, define v( J) := p,( U Z,). Then v is a positive measure on
,EJ
I.P(I) with V({L}) = 0 for every LEI (as has just been shown). But I is, by
hypothesis, a weak Ulam set. Thus p,(Z) = v(I) = 0, showing that Z is a weak
Ulam set.)
() Let X be a set and p, a positive two-valued measure on I.P(X). Let Y be
a weak Ulam set and (AY)YEY a family from I.P(X) such that p,(Ay) = 0
for every y E Y. Then p,( U Ay) = O. yEY
(First assume that (AY)YEY is a disjoint family, and consider the positive
measure
v : I.P(Y) ~ IR, B f---+ p, (U A y ) .
yEB
For the proof of the general case, construct a disjoint family (BY)YEY of
subsets of X such that U By = U Ay.) yEY yEY
('17) The analogous assertions to (13) - (() hold for Ulam sets.
(19) If X is an Ulam set with cardinality Net, then every set with cardinality
Net+! is an Ulam set.
(For every ordinal " define A"f := {f3 I 13 an ordinal, 13 < ,}. A"f and , always have the same cardinality. Denote by , the first ordinal with the
cardinality of Net+! . Then for every 13 < "
A"f = {f3} U AiJ U {a E A"f I 13 < a} .
2.6 Positive Measures 369
Let f..t be a positive measure on ~(A"() such that f..t( {,B}) = 0 for each ,B E A"( .
For each ,B E A"( , let ({)(j : A(j --+ X be an injective mapping. For ,B E A"( and
x E X, define A(,B,x) := {a E A"( I ,B < a, ({)o(,B) = x}. Then for each
x EX, (A(,B, X))(jEA"( is a disjoint family of subsets of A"(. Consequently for
each x EX, {A(,B,x) I f..t(A(,B,x)) > O} is countable. Thus
{(,B, x) E A"( x X I f..t(A(,B,x» > O}
has a cardinal strictly smaller than No+l . Hence, for some fj E A,,(, f..t(A(fj, x» =
o for every x EX. Thus f..t( U A (fj, x» = 0 since X is an Ulam set. But A,ij xEX
is also an Ulam set, and we conclude
f..t(A"() = f..t( {fj}) + f..t(A,ij) + f..t( {,B E A"( I fj < ,B}) ::;
(L) Let X be a weak Ulam set and f..t a strictly positive measure on ~(X)
such that f..t({x}) = 0 for every x EX. Then for each cEnt, c > 0,
there is a subset A of X with 0 < f..t(A) < c.
(Using Zorn's Lemma, there is a filter ~ on X, maximal with respect to
the property that f..t(F) > 0 whenever F E ~. Prove that for every C C X
either C E ~ or X\C E ~. Hence ~ is an ultrafilter. But then there is a
decreasing sequence (Fn)nEIN from ~ such that n Fn = 0.) nEIN
(K) If nt is an Ulam set, then so is every weak Ulam set.
(Let X be a weak Ulam set and f..t a positive measure on ~(X) such that
f..t({x}) = 0 for every x EX. Using (L), construct a sequence (fn)nEIN of
mappings fn : INn --+ ~(X) recursively such that for each n E IN:
i) (fn(k)hElNn is a disjoint family with U fn(k) = X; kElNn
ii) f..t(fn(k)) < 1/2n whenever k E INn;
iii) fn+l(k,j) C fn(k) whenever k E INn and j E IN.
For every 9 E ININ , define
Ag := n fn(g(l), g(2), ... , g(n» . nEIN
370 2. Elementary Integration Theory
Then Il(Ag) = 0 whenever 9 E 1NIN. But X = U Ag, and (1]) implies our 9EIN IN
assertion. )
(A) If we assume the Continuum Hypothesis (i.e., that 2No = Nd , then weak
Ulam sets and Ulam sets coincide.
2.7 Closure, Completion, and Integrals for Positive Measure Spaces 371
2.7 Closure, Completion, and Integrals
for Positive Measure Spaces
Our first goal is to show that every positive measure space has a smallest closed
extension.
Proposition 2.7.1 Let J-L be a positive measure on the set-ring ~ and
let Xl> X 2 be sets containing X(~). For k = 1,2, let (Xk, £k, £k) denote
the Daniell space associated with the positive measure space (Xk'~' J-L) and
(Xk, ~k' J-Lk) the closed positive measure space induced by (Xk, £(£k)' £k)' Then
we have the following.
a) U =I- O} C X(~) whenever f E £(£1) U £(£2) .
b) (~1' J-Ld = (~2' J-L2).
Proof. a) Take k E {1,2} and f E £(£k)' We may assume that f ~ O. There
is an increasing sequence (fn)nEIN in £k with f::; V fn. By Proposition nEIN
2.5.12 a), Un> O} E ~ for every n E IN. Thus
U =I- O} C U Un > O} C X(~) . nEIN
b) Given f E £(£d, define
{ f(x)
x~ 0
if x E X(~)
otherwise.
Note that
for every A E ~. Thus, using a), it is easily verified that the mapping
is an isomorphism of ordered sets such that
for every f E £(£1) . The claim now follows. o
372 2. Elementary Integration Theory
Proposition 2.7.2 Let p, be a positive measure on the set-ring ~. Let
(X(~), ~(p,), J1) denote the closed positive measure space induced by the closure
of the Daniell space associated with (X(~),~, p,). Then for each X :::) X(~) ,
(X, ~(p,), J1) is the smallest closed positive measure space extending (X,~, p,) .
Proof Let X :::) X(~), and let (X, £, i) be the Daniell space associ
ated with the positive measure space (X,~, p,). According to Proposition 1,
(X, ~(p,), J1) is the closed positive measure space induced by (X, C(f), f) . Ev
idently (X,~, p,) ~ (X,~, (p,), J1). Now let (X, !R', p,') be an arbitrary closed
positive measure space for which (~, p,) ~ (~', IL') . By Theorem 2.6.20, there
is a unique closed Daniell space (X, £', i') with the Stone property and such
that (X,~', p,') is the positive measure space induced by (X, C', £') . It follows
that
(X, £, f) ~ (X, £', £')
and therefore
(X,C(f),f) ~ (X,C',f').
Hence
o
We are now ready for the following definition.
Definition 2.7.3 (Measure-space closure) Let p, be a positive measure on
the set-ring !R. Then (X(~), 9l(p,) , J1) denotes the closed positive measure
space induced by the closure of the Daniell space associated with (X(!R),~, p,). We call Ji the closure of p, and (~(p,), J1) the closure of (~, p,). For each
X :::) X(!R) , we call (X, ~(p,), Ji) the closure of (X,!R, p,) . 0
Corollary 2.7.4 For every positive measure space (X,~, p,) , if (X, 6, /I) is a
positive measure space extending (X,~, p,) and if 6 C ~(p,)<T' then 6 C ~(p,)
and
/I = Jile.
In particular, Ji and JiI!Jt6 are the only positive measures on ~(p,) and ~6,
respectively, that extend p,.
2.7 Closure, Completion, and Integrals for Positive Measure Spaces 373
Proof We have
(X, 9l, /-l) ~ (X, <5, v) ~ (X, 9l(v), v)
and (X, 9l(v), v) is closed. It follows that
(X, 9l(/-l), p;) ~ (X, 9l(v), v) .
In particular,
Take A E <5. By hypothesis, A belongs to 9l(/-l)u. According to Propositions
2.6.19 and 2.5.9, we may choose an increasing sequence (An)nEIN from 9l(/-l)
whose union is A. Using nullcontinuity, we see that
sup M(An) = sup v(An) = v(A) < 00. nEIN nEIN
Hence A belongs to 9l(/-l) and v(A) = v(A) = M(A) . o
Some authors require the domains for positive measures to be a-rings. But
in view of Corollary 4, allowing arbitrary set-rings as domains does not expand
the collection of closed positive measure spaces. What is gained is greater ease
in defining a particular measure with which one wants to work.
We now turn our attention to extensions of measures induced by integrals.
Definition 2.7.5 (Measure-space integral and completion) Given a posi
tive measure space (X, 9l, /-l), let (X, £(X, /-l), lx,/J) be the Daniell space asso
ciated with (X, 9l, /-l). Then
For f E £1 (X, /-l), we define
1 f:= r fd/-l:= r f(x)d/-l(x):= r f x,/J Jx Jx Jtx.~
and call Ixfd/-l the /-l-integral of f. For A E £(X, /-l), we define
374 2. Elementary Integration Theory
and call J-tx (A) the J-t-measure of A in X.
Functions belonging to £l(X, J-t) are called J-t-integrable junctions on X
and functions belonging to N(X, J-t) are called J-t-null junctions on X. Sim
ilarly, sets belonging to £(X, J-t) are called J-t-integrable sets in X or J-t
integrable subsets of X and sets belonging to lJ1(X, J-t) are called J-t-null sets
in X or J-t-null subsets of X .
The integral (X,£l(X,J-t),Jx,/-J is called the integral for (X,~,J-t) or the
integral on X associated with J-t.
The closed positive measure space (X, £(X, J-t), J-tX) is called the comple
tion of (X,~, J-t) and J-tx is called the completion of J-t on X or simply the
completion of J-t. If (X,~, J-t) = (X, £(X, J-t), J-tX ), then we call the measure
space (X,~, J-t) complete and we say that J-t is a complete measure on X.
We call hi -sequences J-t-sequences, that is, a monotone sequence (fn)nEIN x." from £l(X, J-t) for which Ux fndJ-t)nEIN is bounded in 1R is a J-t-sequence.
Monotone sequences (An)nEIN from £(X,J-t) for which (J-tX(An))nEIN is bounded
in 1R will also be called J-t-sequences.
Finally, a property P that refers to elements of X is said to hold J-t-almost
everywhere (J-t-a.e.) iff it holds £x,/J-a.e. We write P J-t-a.e. and P(x) J-t
a.e., respectively, for P £x,/J-a.e. and P(x) £x,/J-a.e. 0
Theorem 2.7.6 (Main Theorem on Measure-Space Completion and In
tegral) Let (X,~, J-t) be a positive measure space and (X, £, £) its associated
Daniell space. Then we have the following.
a) (X, £l(X, J-t), Ix) is a closed Daniell space with the Stone property ex
tending (X, £, £) .
b) (X, £(X, J-t)' J-tX) is a closed positive measure space extending (X,~, J-t) .
c)
-x -= {j E 1R I feA E N(£(£)), V A E ~}
-x = {j E 1R I feA E N(X,J-t), VA E~}.
2.7 Closure, Completion, and Integrals for Positive Measure Spaces 375
d) IJl(X, Jl) = 1Jl(£l(X, Jl))
= {B E £(X,Jl) I JlX(B) = O}
= {B C X IBn A E 1Jl(£(£)), V A E 9l}
= {B C X IBn A E IJl(X, Jl), V A E 9l}
= {B C X I JlX(B n A) = 0, V A E 9l}.
e) If f belongs to £l(X, Jl) , then there are disjoint sets Band C such that
f = feB + fee, BE !R(£(£)), C E IJl(X,Jl).
Moreover, feB belongs to £(£) for every B in !R(£(£)).
f) A set A belongs to £(X, Jl) iff there are disjoint sets Band C such that
If A belongs to £(X, Jl), then An B belongs to 9l(Jl) for every B in
9l(Jl) .
g) If f belongs to £1 (X, Jl), then there are a sequence (An)nEIN from 91
and a set B belonging to IJl(X, Jl) such that
{J # O} C (U An) U B. nEIN
The sequence (An)nEIN can be chosen to be increasing or disjoint.
h) If A belongs to £(X, Jl), then there are a sequence (An)nEIN from 91 and
a set B belonging to IJl(X, Jl) such that
A C (U An) uB. nEIN
The sequence (An)nEIN can be chosen to be increasing or disjoint.
i) For every f belonging to £l(X, Jl) ,
r fdJl = sup r feAdJl + inf r feAdJl. ix AE<Jtix AE<Jtix
For every positive f in £ 1 (X, Jl) ,
r fdJl = sup r feAdJl. i x AE<Jt i X
376 2. Elementary Integration Theory
j) For every B belonging to £(X, p,) ,
p,X (B) = sup Jlx (A n B) . AE9t
k) For every B eX, B E IJ1(X, p,) n !Jl(p,) iff for every c > 0 there is a
disjoint sequence (An)nEIN from !)l such that
Be U An and LJl(An) < c. nEIN nEIN
f) For every B eX, B E IJ1(X, p,) iff for every A E !Jl and every c > 0
there is a sequence (An)nEIN from !)l such that
An B c U An and L Jl(An) < c. nEIN nEIN
Proof. Note that 6i(C) =!)l in this instance.
a) follows from Theorem 2.4.7 and Proposition 2.5.18.
b) is a consequence of a) and Theorem 2.6.20 b) => a) .
c) ,d) can be deduced from Propositions 2.2.11, 2.4.4 d) and 2.4.15 d).
e) - h) follow from Proposition 2.4.15.
i),j) are consequences of Proposition 2.4.16.
k) Suppose that BE IJ1(X, Jl) n !)l(Jl) and take c > O. There is an f E Ct
with f ;::: eB and ft(f) < 10/3. Moreover, there is an increasing sequence
(fn)nEIN in C+ whose supremum is f. Given n E 1N, put Bn := Un ;::: 1/2} .
Then (Bn)nEIN is an increasing sequence in !Jl such that Be U Bn and since nEIN
eBn ::; 2fn ::; 2f, we see that Jl(Bn) ::; 2 Ix fdJl < ~c for every n E 1N .
Define Al := Bl and An := Bn \Bn- I whenever n > 1. Then U An = nEIN
n U Bn. Moreover, since P,(Bn) = L p,(Ak) for every n E 1N ,
nEIN k=1
For the converse, take c > 0 and let (An)nEIN be a sequence in !)l satisfying
the conditions stated in k). Then (0, e U An) is an c-bracket for eB relative nEIN
to f. Since c is arbitrary, eB E C(f) and p,(B) = f(eB) = O.
1) In view of f), An B E IJ1(X, Jl) n !)l(Jl) whenever B E IJ1(X, Jl) and
A E !)l. Hence 1) follows from d) and k). o
2.7 Closure, Completion, and Integrals for Positive Measure Spaces 377
Proposition 2.7.7 Let J-l be a positive measure on the set-ring ryt, and take
sets X, Y with
X J Y J X (ryt) .
Then we have the following.
a) The set X\Y belongs to IJl(X, J-l) .
b) IJl(X, J-l) = {A c X I AnY E IJl(Y, J-l)}.
c) N(X, J-l) = {J E lIr' I fly E N(y, J-l)} .
d) £(X, J-l) = {A c X I AnY E £(Y, J-l)}.
e) .c l (X, J-l) = {J E lRx I fly E .cl(y, J-l)} .
f) For every A E £(X, J-l) , J-lx (A) = J-lY (A n Y) .
g) For every f E .c l (X, J-l) , JxfdJ-l = Jy flydJ-l.
Proof. a) For every A in ryt,(X\Y)nA=0.ByTheorem6d), X\Y belongs
to IJl(X,J-l).
b) We write (X,.c(X),£x) and (Y,.c(Y),£y) for the Daniell spaces asso
ciated with the positive measure spaces (X, ryt, J-l) and (V, ryt, J-l) , respectively.
Take A E IJl(X, /1) . In other words, suppose that
An BE 1Jl(£(£x))
for each B in ryt. From Proposition 1 and Theorem 6 d), we have
1Jl(£( £ x)) = 1Jl(£( £y)) c '-P(Y) .
It follows that
(A n Y) n BE 1Jl(£(£y)) (1)
for each B in ryt, hence that AnY belongs to IJl(Y, It) . Suppose, conversely,
that A is a subset of X whose intersection with Y belongs to IJl(Y, /1) . Then
(1) holds for each B in ryt, and AnY must belong to IJl(X, It) . Since A\Y
also belongs to IJl(X, It) , by a), we conclude that A belongs to IJl(X, It).
c) Assertion c) follows from b) .
d).f) Take A E '£(X, /1) . According to Theorem 6 f), there are disjoint sets
B E ryt(fl) and C E IJl(X, J-l) such that A = B u C. By b), en Y belongs
378 2. Elementary Integration Theory
to !J1(Y, f.t) . Invoking Theorem 6 f) again, we conclude that AnY belongs to
£(Y, f.t) . Moreover,
The argument in the other direction is similar.
e),g) Take IE £I(X, f.t)+. By Proposition 2.5.21 b), there is an increasing
sequence (fn)nEIN of positive £(X, f.t)-step functions such that I = V In. In nEIN
view of d), (fnIY)nEIN is an increasing sequence of £(Y, f.t)-step functions with
supremum Ily. In light of f),
sup r Inlydf.t = sup r Indf.t = r fdf.t < 00. nEINiy nEINix ix
Hence
(2)
For an arbitrary I E £1 (X, f.t) , (2) follows by decomposing I as 1= r - 1- . The argument in the other direction is similar. o
Part a) of the preceding proposition shows that £(X, f.t) differs from
£(Y,f.t) , and £l(X,f.t) differs from £l(y,f.t) , unless X = Y. Thus there is
good reason for including X in the denotations for the various objects con
nected with measures.
We now turn to the problem of finding sufficient conditions for two positive
measure spaces to generate the same integral.
Proposition 2.7.8 Let (X, 9t, f.t) and (X, 6, 1/) be positive measure spaces
with
Suppose that lor every set A in 6 there is a countable lamily (A')'EI from 9t
such that
A\UA, E!J1(X,I/). (3) ,EI
Then
and
2.7 Closure, Completion, and Integrals for Positive Measure Spaces 379
Proof. Let (X, £, f) be the Daniell space associated with (X, 9l, /1) . The hy
potheses imply that
and consequently that
(X,£(f)'£):s (X,£l(X,v),l ) x,v
(4)
Now we use Theorem 6 d) to show that
91(X, /1) C 91(X, v) . (5)
Take B E 91(X, /1) . Given A E 6, let (A.)LEI be a countable family from 9l
for which (3) holds. For every tEl, B n AL belongs to 91(£( t')) and hence, in
view of (4), to 91(X, v). Thus U (B n A L ) E 91(X, v). Since LEI
AnB c (U(B n A.)) U (A\UA) LEI LEI
we conclude that An B E 91(X, v) . By Theorem 6 d), B E 91(X, v) .
Now take f E £l(X, /1). Then f = 9 /1-a.e. for some 9 E £(f). By (5),
f = 9 v-a.e. In view of (4), f E £l(X,V) and
lfdv = 19dV = Z(g) = lfd/1,
proving the first assertion of the proposition. The second assertion follows from
the first. D
Corollary 2.7.9 Let (X, 9l, /1) and (X, 6, v) be positive measure spaces. If
and
then (X, 9l, /1) and (X, 6, v) have the same integral and the same completion:
380 2. Elementary Integration Theory
Proof. Take A E 6. By hypothesis, A belongs to £(X, p,). Theorem 6 h)
ensures the existence of a countable family (AL)LEI from ~ such that
A\UAL E 91(X,p,). LEI
In order to apply Proposition 8, we show that
A\UAL E 91(X,v). (6) LEI
Since ~ C £(X, v) and £(X, v) is a 6-ring (Proposition 2.6.19), the set
A\ U A L, which can be written as n (A\A) , must belong to £(X, v). Ac-LEI LEI
cording to Theorem 6 f), there are disjoint sets B in ~(v) and C in 91(X, v)
such that
A\UA=BUC. LEI
Corollary 4 implies that
from which we conclude that B belongs to £(X, p,) and ILX (B) = YJ(B) . Since
B is a subset of A \ U A , which is a p,-null subset of X, B must belong to LEI
91(X, v) . Consequently, (6) holds.
Applying Proposition 8, we conclude that
Reyersing the argument, we obtain the opposite inequality and the corollary
follows. 0
Corollary 2.7.10 Let (X,~, p,) and (X, 6, v) be positive measure spaces. If
~ C 6,s C £(X,IL)
and
XI X p, 6,\ = V 16.
then (X,~, p,) and (X, 6, v) have the same integral and the same completion.
2.7 Closure, Completion, and Integrals for Positive Measure Spaces 381
and
Using Corollary 9, we conclude that
On the other hand,
(6, /.I) ~ (£(X, v), VX),
Using Corollary 9 again, we have
(X, £(X, /.I), /.IX) = (X, £(X, v), VX) .
The corollary follows. o
A minor consequence is the idem potence of measure-space integrals and
completions.
Corollary 2.7.11 For every positive measure space (X,~,J.l)
Proof. Apply Corollary 10 using (X, 6, /.I) := (X, £(X, J.l)' J.lX) . o
We now investigate consequences arising for measure-space integrals when
the underlying measure space satisfies various boundedness conditions.
382 2. Elementary Integration Theory
Proposition 2.7.12 Let (X,~,J,t) be a positive measure space and (X,.c,l)
its associated Daniell space. If the pair (X,~) is a-finite, then the following
assertions hold.
a) (X, .cl(X, J,t), Ix) = (X, :e(l), £) .
b) (X, £(X, J,t), J,tX) = (X, ~(J,t), JL) .
Proof By hypothesis, there is a countable family (A')'EI from ~ whose union
is X. The functions eA, belong to .c and
In other words, the Riesz lattice .c is a-finite. We conclude a) from Corollary
2.4.14, and we conclude b) from a) . 0
Proposition 2.7.13 Let (X,~, J,t) be a positive measure space. The following
are equival::nt.
a) (X,~, J,t) is a-bounded.
b) There is a countable family (A),E/ from ~ such that X\ U A E 'EI
IJl(X, J,t) .
c) X belongs to !R(e l (X, J,t)) .
d) The pair (X, £(X, J,t)) is a-finite.
Proof a)::::} b). Given A E ~, there is a countable family (A,),E/ from ~
with
inf J,t(A \A) = O. 'EI
Then
Theorem 6 d) now shows that X\ U A, E IJl(X, J,t). 'EI
b) ::::} d) is trivial.
d) ::::} c) follows from Proposition 2.4.2 e) .
2.7 Closure, Completion, and Integrals for Positive Measure Spaces 383
c) =* a) . By Theorem 6 g), there are an increasing sequence (An)nEIN from
~ and a set B belonging to IJt{X, J.t) such that
x = (u A~) UB. nEIN
It follows that A \ U An belongs to IJt{X, J.t) for every A in ~ and that nEIN
o
Bounded measures play a very special role. The completions of such mea
sures provide a new kind of set system.
Definition 2.7.14 For ~ C ~(X), ~ is said to be a a-algebra on X iff
~ is a a-ring and X belongs to ~. 0
Thus a-algebras are only defined relative to some underlying set X. They
can be characterized in various ways.
Proposition 2.7.15 Let ~ be a set of sets and take X :) X(~). Then the
following are equivalent.
a) ~ is a a-algebra on X.
b) ~ is a 8 -ring and X belongs to ~.
c) The empty set belongs to ~. X\A belongs to ~ for every A in ~ and
U An belongs to ~ for every sequence (An)nEIN from ~. nEIN
Proof. We leave the details to the reader. Observe that
A\B = X\«X\A) U B) . o
Proposition 2.7.16 Let (X,~, J.t) be a positive measure space. Then the fol
Proof. a) ~ b). By Corollary 2.6.8 and Proposition 13 a) ~ c), X E
6t(£l(X,J-t)). According to Theorem 6 g), there are an increasing sequence
(An)nEIN from 9t and a set BE 'J1(X, J-t) such that
X = (U An) uB. nEIN
Since
U An belongs to £(X, J-t). Thus X E £(X, J-t). nEIN
The implications b) ~ c) ~ d) ~ a) and b) ~ e) are left to the reader.
o
Example 2.7.17 Let 9t be a set-ring on X and x E X such that {x} E 9t.
The reader can easily verify the following assertions about the Dirac measure
§x := §~ concentrated in x.
-x N(§x) = {J E lR I f(x) = D},
-x £l(§x) = {J E lR I f(x) E lR},
f fd§x = f(x) whenever f E £1 (§x) . o
Given a Daniell space (X, £, £), we finally pursue the question of when
(X,£l(£),it) is generated by a positive measure space.
2.7 Closure, Completion, and Integrals for Positive Measure Spaces 385
Theorem 2.7.18 Let (X,C,e) be a Daniell space such that C1 (e) has the
Stone property. Denote by (X, 9't, /.1) the positive measure space induced by
(X, C 1 (e), L). Then (X, 9't, /.1) is complete and
(7)
Proof Let (X, C, e') be the Daniell space associated with (X, 9't, /.1) . By virtue
of Theorem 2.6.20 b) :::} a), (X, 9't, /.1) is closed and
(X,£(t),£i) = (x,c 1 (e), 1) (8)
Thus (7) follows if we can show that I)1(X, /.1) C l)1(e) . So take B E I)1(X, /.1)
and A E iR(£(e)). By (8), A E iR(£(e')). Then, using Theorem 6 e),
B n A E 1)1(£(t)) = l)1(e)
and hence B n A E 1)1(£(e)). It follows that BE l)1(e).
Finally, the completeness of (X, 9't, /.1) is a consequence of (7). 0
Corollary 2.7.19 Let (X, 12, e) be a Daniell space. Then the following are
equivalent.
a) There is a positive measure space (X, 9't, /.1) such that
b) 121 (e) has the Stone properly. o
Observe that the Daniell space (X, C1(e), Ie) from Example 2.4.10 lacks the
Stone property and is therefore not generated by a positive measure space.
386 2. Elementary Integration Theory
2.8 Measurable Spaces and Measurability
Measurability plays several roles in the theory of integration. First, it yields an
integrability criterion, in fact the most important integrability criterion, which we treat in Section 2.9. It turns out that measurability, when properly defined,
provides a necessary, though not sufficient, condition for integrability. Thus
the set of measurable functions is larger than the set of integrable functions.
Measurable functions, it further turns out, form a class that is large enough
to permit all appropriate limit operations as well as algebraic operations, a
property that makes this class very important in the theory of integration. One
can place in this framework, for instance, the entire theory of .cP-spaces which
will be treated in the next chapter.
Definition 2.8.1 For!Jt a 6-ring on a set X, we define
9Jt(X,!Jt) := {A c X I An BE !Jt, VB E!Jt},
-x M(X,!Jt) := {J E lR I {J < a} E 9Jt(X,!Jt), Va E lR},
and we say that X is a measurable space with 6-ring !Jt or that (X,!Jt)
is a measurable space. Sets belonging to 9Jt(X,!Jt) are called !Jt-measurable
subsets or measurable sets of X . Functions belonging to M(X,!Jt) are called
!Jt-measurable junctions on X. o
Proposition 2.8.2 For every measurable space X with 6-ring !Jt, the set
9Jt(X,!Jt) contains !Jt and is a a-algebra on X. If X belongs to !Jt, then
9Jt(X,!Jt) = !Jt.
Proof. That!Jt c 9Jt(X,!Jt) is evident, as is the reverse inclusion when X
itself belongs to !Jt. Certainly, 0 E 9Jt(X,!Jt) . If A E 9Jt(X,!Jt) , then for every
BE!Jt
(X\A) n B = B\(A n B) E !Jt,
that is, X\A E 9Jt(X,!Jt) . Finally, let (An)nEIN be a sequence from 9Jt(X,!Jt).
Note for B E !Jt that
Using Proposition 2.5.2 e), we conclude that U An belongs to 9Jt(X,!Jt). By nEIN
Proposition 2.7.15 c) ::::} a), 9Jt(X,!Jt) is a a-algebra on X. 0
2.8 Measurable Spaces and Measurability 387
Proposition 2.8.3 Let X be a measurable space with 8 ~ring 9t. Then the
following assertions hold.
a) For A eX, A is an 9t~measurable subset of X iff eA zs an 9t~
measurable function on X.
b) Constant extended~real functions on X are 9l.~measurable.
c) For every 9l.~measurable function f on X, the sets
{j=oo}, {j = -oo}, {x E X I f(x) is real}
are 9t~measurable.
d) If (A')'EI is a countable disjoint family of 9t~measurable subsets of X
whose union is X and if (1,),0 is a family of 9l.~measurable functions
on X , then the function
f: X --t IR, X f----t f,(x) (x E A" ~ E 1)
is 9t ~measumble.
e) For every 9l.~measurable function f on X, the function
h : X --t IR, X f----t
is 9t~measurable.
{ f(x)
o if f(x) E IR
if f(x) 1. IR
f) For every real number a and for all 9t~measurable functions f and 9
on X , the sets
{j<g+a}, {j::;g+a}, {j=g+a}, {j#g+a}
are 9t ~measurable.
Proof. a) For a E IR,
if 1 < a
if 0 < a ::; 1
if a ::; O.
In view of Proposition 2, the assertion follows.
388 2. Elementary Integration Theory
b) is trivial.
c) Since
U = oo} = X\ U U < n}, nElN
U = -oo} = nU < -n}, nElN
{x E X I f(x) is real} = X\(U = oo} U U = -oo})
-x for every f E IR ,c) follows from the fact that 9Jt(X,!R) is a a-algebra on
X. d) For each real number a,
U < a} = U(U. < a} n A.) . • EI
Since 9Jt(X,!R) is a a-algebra, the assertion follows.
e) Assertion e) follows from b),c), and d).
f) To verify that U < 9 + a} is !R-measurable, it suffices to note, in view
of Proposition 2, that
U < 9 + a} = U (U < t3} \ {g < t3 - a} ) . .8EQ
Indeed, given x EX, f(x) < g(x) + a iff there is a rational number t3 such
that f(x) < t3 ::; g(x) + a. The !R-measurability of the other three sets now
follows again in view of Proposition 2, since
U :5 9 + a} = X\ {g < f - a},
U = 9 + a} = U ::; 9 + a}\{f < 9 + a},
U f. 9 + o} = X\U = 9 + o}.
Theorem 2.8.4 Let X be a measurable space with a-ring !R.
o
a) If f is an !R-measurable function on X, then so is of, whenever 0 is
any extended-real number.
b) If f and 9 are !R-measurable functions on X and if f + 9 is defined,
then f + 9 is !R-measurable.
c) For every countable family (j')'EI of !R-measurable functions on X, the
functions
2.8 Measurable Spaces and Measurability 389
/\f. and V f • • EI .EI
are both !R-measurable.
-x d) For every f E IR , f is £R-measurable iff both f+ and f- are £R-
measurable.
e) If f is an £R-measurable function on X, then so is If I·
f) If f and 9 are !R -measurable functions on X , then so is f 9 .
g) If f is an £R-measurable function on X and
{ _1 if f(x) E IR\{a}
9 : X --t IR, X t----t al(X)
otherwise,
then 9 is !R-measurable.
h) For every sequence (fn)nEIN of !R-measurable functions on X, the func
tions
lim sup fn and lim inf fn n-+oo n-+oo
are both !R-measurable.
i) If (fn)nEIN is a sequence of £R-measurable functions on X which is order
convergent in IRx , then the function lim fn is £R-measurable. n-+oo
j) For every countable family (f')'EI of £R-measurable real functions on X,
if (f.).EI is summable, then E f. is £R-measurable . • EI
k) Every rot(X, !R)-step function on X is !R-measurable.
Proof. a) If Q E IR, distinguish the cases Q = a, Q > a, Q < a. If Q = 00,
note that for each {3 E IR
{oof < {3}= {{J~a} {J < a}
if {3 > a
if (3 ~ a.
In view of Proposition 3 b),f), oof is £R-measurable. Finally,
{-oof < {3} = {-{3 < oo!}
390 2. Elementary Integration Theory
for every real number f3, so -oof is also 9{-measurable.
b) If f + 9 is defined, then
U+g < a} = U < -g+a}
for every real number a. Thus b) follows from a) by Proposition 3 b),f).
c) For every real number a,
{ /\ ft < a} = UUt < a}. tEl tEl
Since 9J1(X,9{) is a a-algebra on X, we conclude that /\ ft is 9{-measurable. tEl
Since
V ft = -/\(-ft) tEl tEl
it follows from a) that V ft is also 9{-measurable. tEl
d),e) These assertions follow from a) - c).
f) Suppose first that f, 9 ~ 0 and take a E 1R. If a :::; 0, then U 9 < a}
is empty. So suppose that a > O. We claim that
That ~ holds is evident. Suppose that x fails to belong to the set on the right
hand side. Then f(x) > 0, and for every strictly positive rational number f3
we have
x ~ {g < f3} n {f < ~} .
If g(x) = 00, then f(x)g(x) = 00 and hence x ~ Ug < a}. If g(x) < 00,
then f(x) ~ alf3 whenever f3 E Q, /3 > g(x). Thus
f(x)g(x) = inf{f3f(x) I f3 E Q, /3 > g(x)} ~ a,
and so x ~ U 9 < a} , establishing (1). It follows that f 9 is 9{-measurable.
Now let f and 9 be arbitrary 9{.-measurable functions on X. Set
A:= U ~ a}, B:= U < O},
C:= {g ~ O}, D:= {g < O}.
2.8 Measurable Spaces and Measurability 391
The sets A n G, AnD, B n G, B n D are pairwise disjoint, and their
union is X. According to what we have already established, the functions f+ g+ , - f+ g- , - f- g+ , f- g- are all 9\-measurable. Since
rg+ on AnG
-rg- on AnD fg
-f-g+ on BnG
f-g- on BnD
it follows from Proposition 3 d) that fg is 9\-measurable.
g) For a E IR,
{J ~ O}U U < J} if a> 0
{g < a} {J<O}\{J=-oo} ifa=O
{J < O} n U < J}
so g) follows from Proposition 3 b),f) .
h) Assertion h) follows from c).
if a < 0,
i) Assertion i) follows from h) and Proposition 1.8.23 b).
j) If I is empty, the assertion is trivial. If I is finite and nonempty,
the assertion follows from b) by complete induction. Suppose finally that I
is countably infinite. It suffices to prove the assertion for a summable sequence m
(fn)nEIN of 9\-measurable real functions on X. For such a sequence, E fk is k=l
9\-measurable (m E IN) , and
m
~ fn = lim ~ fk L...J m-+oo L.. nEIN k=l
(Proposition 2.3.28 a)). j) is now a consequence of i).
k) follows from a) and j), together with Proposition 3 a). o
We stress that M(X,9\) is not, in general, a Riesz lattice. To wit, the
constant function ooex is always 9\-measurable, so M(X,9\) cannot be a
Riesz lattice unless M(X,9\) = IRx , which is seldom the case.
Proposition 2.8.5 Let X be a measurable space with 8-ring 9\ and take
f in M(X,9\). Then there is a sequence (fn)nEIN of rot(X, 9\)-step functions
on X such that lim fn = f and Ifni ~ If I for every n E IN. If f ~ 0, the n-+oo
sequence (fn)nEIN can be chosen increasing, with fn ~ 0 for every n E IN .
392 2. Elementary Integration Theory
Proof For I E M(X, 9l)+, the assertion follows from Proposition 2.5.19.
If I is an arbitrary 9l-measurable function on X, then 1+ and 1- belong
to M(X, 9l)+. Choose increasing sequences (9n)nEIN and (hn)nEIN of positive
9Jl(X, 9l)-step functions whose suprema are 1+ and 1- , respectively. Set In :=
9n - hn . Then the sequence (fn)nEIN has the required properties. 0
2.9 Measurability versus Integrability 393
2.9 Measurability versus Integrability
In the preceding section, measurability relative to a o-ring was defined and
investigated. It was exactly the defining property of o-rings that implied the
important properties formulated in Theorem 2.8.4: measurability relative to a
ring of sets would have been to weak. Given a positive measure space (X,~, /1-) ,
then, it is not be particularly useful to speak of measurability relative to ~.
Instead, we shall use either the o-ring ~6 generated by ~ or the o-ring
.c(X, /1-) of /1--integrable subsets of X.
The first part of the present section discusses /1--measurable objects and
their relation to /1--integrable objects. Upper and lower /1--integrals, and inner
and outer /1--measures, which are introduced in Definition 4, provide useful tools
for describing the relationship between measurability and integrability.
Definition 2.9.1 Let (X,~, /1-) be a positive measure space. Then
VJl(X, /1-) := VJl(X, .c(X, /1-)) ,
M(X, /1-) := M(X, .c(X, /1-)) .
Sets belonging to VJl(X, /1-) and functions belonging to M(X, /1-) are called /1-
measurable on X . 0
Proposition 2.9.2 Let (X,~, /1-) be a positive measure space. Then the fol
lowing assertions hold.
a) 'J1(X, /1-) C .c(X, /1-) C 6t(.c1 (X, /1-)) C VJl(X, /1-) .
b) If A E .c(X, /1-) , BE VJl(X, /1-) and Be A, then B belongs to .c(X, /1-) .
c) If B is a subset of X for which
A~B E 'J1(X, /1-)
for some A E VJl(X, /1-), then B belongs to VJl(X, /1-) .
d) N(X, /1-) C .c1(X, /1-) C M(X, /1-).
e) If g is an extended-real function on X that is /1--a. e. equal to a /1-
measurable function f on X, then g is /1--measurable.
394 2. Elementary Integration Theory
I) The conditions I,g E M(X,J-l) , hE m.x,
h(x) = I(x) + g(x) J-l-a.e.
imply that h belongs to M(X, J-l) .
Proof a) The first two inclusions are already known. The third is a consequence
of Propositions 2.5.21 c) and 2.8.2.
b) is evident.
c) The hypothesis implies that both B\A and A\B are J-l-measurable.
S~nce
B = (A\(A\B)) U (B\A) ,
B is also J-l-measurable (Proposition 2.8.2).
d) The first inclusion is already known. For the second, since 1 is J-lmeasurable if 1+ and 1- are (Theorem 2.8.4 d)), it suffices to show that
(1)
By Proposition 2.5.21 b), every function in .c1(X, J-l)+ is the supremum of a
sequence of .c(X, J-l)-step functions, which implies that (1) holds (Theorem 2.8.4
c),k)).
e) With 1 and 9 as hypothesized, set
A:={f#g}.
The sets A and X\A both belong to rot(X, J-l) and the function leX\A belongs
to M(X, J-l) (Proposition 2.8.3 a), Theorem 2.8.4 f)). Since geA is J-l-null, it
belongs to .c 1 (X, J-l) and therefore to M (X, J-l) , by d). Then, in view of Theorem
2.8.4 b),
9 = leX\A + geA E M(X, J-l) .
f) follows from e) and Theorem 2.8.4 b).
Corollary 2.9.3 Let (X,!Jl, J-l) be a positive measure space. Then, given 1 E .c1(X,J-l)+ and a E m., a> 0,
{I> a} E .c(X,J-l)' {f ~ a} E .c(X,J-l)
and
o
2.9 Measurability versus Integrability 395
Proof. By Proposition 2.5.21 a), {J ~ a} E £(X,J-t). As a J-t-measurable
subset of this set, {J > a} belongs to £(X,J-t) too (Proposition 2 b),d)).
Moreover,
Definition 2.9.4 Let (X, 9l, J-t) be a positive measure space. For f E lRx ,
we define the upper and lower J-t-integral of f by
1* fdJ-t:= (1 ) * (I) , X X,Jl
r fdJ-t:= (r ) (I), J*x JX,Jl *
respectively. For A eX, we define the outer and inner J-t-measure (C.
Caratheodory, 1914) of A to be, respectively,
J-t*(A) := l* eAdJ-t,
J-t*(A):= r eAdJ-t. J*X o
The rules governing upper and lower integrals were already discussed in
Section 2.3. It goes without saying that analogous rules hold for inner and outer
measures. It will prove useful to record the most important of these results.
Proposition 2.9.5 Let (X, 9l, J-t) be a positive measure space.
a) For every subset A of X ,
b) If A c B eX, then
396 2. Elementary Integration Theory
c) For every nonempty, countable, directed upward family (A.),El from
~(X),
p,* (UA,) = supp,*(A,). ,EI 'EI
d) For every countable family (A')'EI from ~(X),
p,* (UA.) ~ 2:* p,*(A,). ,EI LEI
e) For every disjoint family (A.)LEI from ~(X),
f) A subset A of X belongs to .c(X,p,) iff p,*(A) and p,*(A) are real and
equal. If A belongs to £(X, p,), then
g) m(X,p,) = {A c X I p,*(A) = O}.
Proof. For a),b), and c), use Proposition 2.3.17, Propositions 2.3.18 c) and 2.3.15 e), and Theorem 2.3.16 a), respectively. Assertions d) and e) follow from
Proposition 2.3.34. Assertions f) and g) follow from Propositions 2.3.19, 2.3.15
a), and 2.3.18 a). 0
Example 2.9.6 Consider again the measure space (X, 9t, 6x ) ,with {x} E 9t
(Example 2.7.17). Then
r* fd6x = r fd6x = f(x) ix i*X
-x M(X,6x ) = 1R ,
-x whenever f E 1R . o
Example 2.9.7 Let 3" be the set of finite subsets of X, and let X be
counting measure on 3" (Examples 2.6.10, 2.6.17). In view of Example 2.3.22
and Corollary 2.4.12, we have
2.9 Measurability versus Integrability 397
Hence £(X, X) = ~ and therefore
Moreover,
for every A eX.
rot(X, X) = ~(X) , -x
M(X,X) = 1R .
r* * -x JJl fdX = L f(x) for every f E 1R , X xEX
{ X(A)
X*(A) = X*(A) = 00
if A E ~
if A ¢ ~
o
Next we prove the important result that measurable sets have identical inner
and outer measure.
Theorem 2.9.8 For every positive measure space (X,!oR, JJ) , if A belongs to
rot(X, JJ), then
(2)
Proof. Since JJ*(A) $ JJ*(A) , there is nothing to prove if JJ*(A) = 00. Assume therefore that JJ*(A) < 00. In this case we claim that the JJ-measurable set A
must actually be JJ-integrable and therefore satisfy (2). Let
Q := sup{JJx (B') I B' E £(X, JJ), B' c A}
and note that Q is real (Proposition 5 b),f)). Choose an increasing sequence
(Bn)nElN of JJ-integrable subsets of A such that
sup JJx (Bn) = Q.
nElN
Set
B:= U Bn , C:= A\B. nElN
Then B c A, B belongs to £(X, JJ) and JJx (B) = Q. Hence C belongs to
rot(X, JJ). Let D E !oR. Then C n D belongs to £(X, f.l) and
Q = JJX(B) $ f.lX(B) + JJX(Cn D) = JJX(B u (Cn D)) ::; Q,
so JJx (C n D) = 0 and C n D belongs to lJl(X, JJ). Since D was an arbitrary
set in !oR, we conclude that C belongs to lJl(X,JJ) and therefore to £(X,JJ).
As claimed, A belongs to £(X, JJ) and (2) holds. 0
398 2. Elementary Integration Theory
Corollary 2.9.9 For every positive measure space (X, vt, J.L) , if f E VJ1(X, J.L)+ ,
then
(3)
Proof The validity of (3) follows immediately from Theorem 8 if f is the
characteristic function of a J.L-measurable set.
Suppose next that f is a positive VJ1(X, J.L)-step function, say
for some finite family (A)'EI from VJ1(X, J.L) and some family (a,),El from
lR+ . Using Proposition 2.3.34, we have
Since the reverse inequality always holds, (3) holds.
Finally, let f be an arbitrary positive J.L-measurable function on X. Ac
cording to Proposition 2.8.5, there is an increasing sequence (fn)nEIN of positive
VJ1(X, J.L)-step functions whose supremum is f. Using Theorem 2.3.16 a) and
Proposition 2.3.18 c), we have
Once again, (3) holds. o
Our next theorem characterizes J.L-integrable objects within the class of
J.L-measurable objects.
Theorem 2.9.10 Let (X, vt, J.L) be a positive measure space. Then for every
set A C X and for every function f E lRx we have the following.
a) A E £'(X, J.L) iff A E VJ1(X, J.L) and J.L* (A) < 00.
b) f E Cl(X, J.L) iff f E M(X, J.L) and f; IfldJ.L < 00.
Proof a) We have already established that J.L-integrability implies J.L-measur
ability (Proposition 2 a». The rest of a) restates Proposition 5 f) and Theorem
8.
b) If f is J.L-integrable, then so is If I and, by Proposition 2.3.19 a),
2.9 Measurability versus Integrability 399
Ix* IfldJ.t < 00. (4)
Moreover, in view of Proposition 2 d), f is J.t-measurable.
Suppose, conversely, that f is J.t-measurable and satisfies (4). Then both
rand f- are J.t-measurable (Theorem 2.8.4 d)) and
(Proposition 2.3.15 e)). Combining Corollary 9 with Proposition 2.3.19 a), we
conclude that rand f- belong to Cl(X,J.t). Hence f belongs to Cl(X,J.t). D
Corollary 2.9.11 Let (X,!Jt, J.t) be a positive measure space and (X, C, f) its
associated Daniell space. Then the following assertions hold.
a) If f E C1(X, J.t), 9 E M(X, J.t) and
Igl $ If I J.t-a.e.,
then 9 belongs to C1(X, J.t).
b) If f E £(f) , 9 E M(X,J.t) and
Igl $Ifl £(f)-a.e.,
then 9 belongs to £( f) .
c) The product of a bounded J.t-measurable function with a J.t-integrable func
tion is J.t-integrable. In fact, if f belongs to M(X, J.t) and satisfies
oex $ f $ j3ex J.t-a.e.
with o,j3 E JR, then fg belongs to C1(X,J.t) for every 9 E C1(X,J.t) and
satisfies
° Ix gdJ.t $ Ix f gdJ.t $ j3 Ix gdJ.t
for every 9 E Cl(X, J.t)+.
Proof. a) follows from Theorem 10 b).
b) By a), 9 belongs to C1(X,J.t). By Theorem 2.7.6 e), ge{NO} belongs to
£(f) . But Igl $ If I £(f)-a.e. implies that 9 = ge{f;/:O} £ (f)-a.e., so 9 belongs to £(f).
400 2. Elementary Integration Theory
c) is left to the reader. o
We present some equivalent characterizations of tt-measurable objects.
Proposition 2.9.12 Let (X,!>t, tt) be a positive measure space. Then for each
subset A of X , the following are equivalent.
a) A E VJ!(X, tt) .
b) feA E M(X,tt) for every f E M(X,tt)·
c) An B E '£(X, tt) for every B E !>t.
d) feA E £}(X,tt) for every f E £}(X,tt).
Proof a) ~ b) is a consequence of Theorem 2.8.4 f).
b) ~ c). Let BE!>t. Then eAnB = eAeB E M(X,tt) , by b). Now apply
Proposition 2.8.3 a) and Proposition 2 b) .
c) ~d). We use the Induction Principle (Theorem 2.4.17). Let
The identity eBeA = eBnA shows that eB belongs to F if B belongs to !>t.
Consequently, F contains all !>t-step functions on X. Let (gn)nEIN be a ttsequence from F and g an extended-real function on X such that
lim gn = g tt-a.e. n-+oo
Note that g must belong to £1 (X, tt) . Since
for every n, (gneA)nEIN is a tt-sequence from £l(X, tt) . Moreover,
We conclude that geA belongs to £1(X,tt). In other words, g belongs to F. By the Induction Principle, F = £1(X, tt) .
d) ~ a) is trivial. o
Proposition 2.9.13 Let (X,!>t, tt) be a positive measure space. Then for each
function f E lR! , the following are equivalent.
2.9 Measurability versus Integrability 401
a) f E M(X,Jl).
b) feA E M(X,Jl) for every A E!R.
c) f 1\ aeA E £l(X,Jl) for every A E!R and every a E JR+.
d) . f 1\ neA E £l(X, Jl) for every A E!R and every n E IN .
Proof. a) =} c). Use Theorems 10 b) and 2.8.4 c).
c) =} d) is trivial.
d) =} b). Note that
feA = V (f 1\ neA) nElN
for every A E !R, and apply Theorem 2.8.4 c).
b) =} a). Take a E JR. For each A in !R,
U < a} n A = U e A < a} n A E £( X, Jl) .
A.ccording to Proposition 12 c) =} a), U < a} is Jl-measurable. o
\Ve next pursue the question as to when two positive measure spaces gen
erate the same measurable objects.
Proposition 2.9.14 Let (X,!R, Jl) and (X, 6, v) be positive measure spaces
such that £(X, Jl) C £(X, v) . If for each C E £(X, v) there is aBE £(X, Jl)
with C6.B E SJ1(X, v), then
9J1(X, Jl) C 9J1(X, v) and M(X, Jl) C M(X, v) .
Proof. Take A E 9J1(X, Jl) and C E £(X, v) . Then, choosing B E £(X, Jl) as
hypothesized, the representation
A. n C = ((A n B)\(A n (B\C))) U (A n (C\B))
shows that An C is v-integrable. Since C was arbitrary, A E 9J1(X, v) . The
second inclusion follows from the first. o
One aim of the discussion commenced in Section 2.8 was to expand the
collections of Jl-integrable sets and functions in order to obtain better proper
ties. With Jl-measurable objects, upper and lower Il-integrals, and inner and
402 2. Elementary Integration Theory
outer J.t-measures, that objective is now achieved. Theorems 2.8.4 and 2.9.10
enunciate the desired properties.
But the story continues. Formulating measurability in terms of the mea
surable space (X, 'c(X, J.t))) is not always satisfactory. Difficulties arise, in par
ticular, when we work with different meaSUres. Furthermore, one often needs
relationships to the set-ring used to determine measurability, and in this re
spect 'c(X, J.t) is too large. In these two respects, measurability relative to the
measurable space (X,~) is sometimes very useful. Accordingly, we investigate
9lrmeasurability yet.
Proposition 2.9.15 For every positive measure space (X,!>t, J.t) ,
!JJt(X, !>to) c !JJt(X, J.t) and M(X, !>to) C M(X, J.t) .
Proof. The first inclusion follows from the characterization
!JJt(X, J.t) = {A C X I An BE 'c(X, J.t), VB E !>t}
(Proposition 12 a) - c)), and the second inclusion follows from the first. 0
We investigate the extent to which various kinds of functions from lRx can
be approximated by !>trmeasurable functions.
Theorem 2.9.16 Let (X, 9l, J.t) be a positive measure space and (X, C, £) its associated Daniell space. Then the following assertions hold.
a) For each f E £(l) there are
g, hE M(X, !>to) n £(£)
such that
g~f~h, 9 = h J.t-a.e.
b) For each f E r.l(X,J.t) , there is a
9 E M(X, !>to) n £(£) n IRx
such that
f = 9 J.t-a.e.
2.9 Measurability versus Integrability 403
-x c) For each f E lR ,there is agE M(X, 9t6) such that
g:::; f p,-a.e., r f dp, = r gdp, . i*X i*X
If f*X f dp, is real, then 9 can be chosen from
If there is a countable family from 9t whose union contains {J < O}, then 9 can be chosen so that 9 :::; f .
-x d) For each f E lR there is an hE M(X, 9t6 ) such that
f :::; h p,-a.e., [* fdp, = [* hdp,.
If f; fdp, is real, then h can be chosen from
If there is a countable family from 9t whose union contains {J > O} , then h can be chosen so that f ::; h .
Proof. a) Take f E l( £). There are an increasing sequence (gn)nEIN from
C t and a decreasing sequence (hn)nEIN from Ct such that the sequences
(£t(gn))nEIN and (et(hn))nEIN are real,
V gn ::; f::; 1\ hn , nEIN nEIN
and
Then the functions
g:= V gn, h:= 1\ hn nEIN nEIN
meet the requirements (Theorem 2.8.4 k),c), Propositions 2.3.11 and 2.2.13).
b) Take f E C1(X,p,) and choose l' E l(£) with f = l' p,-a.e. By a),
there is a
g' E M(X, 9t6) n l(£)
404 2. Elementary Integration Theory
with g' = f' p-a.e. Define
{ g'(x) if g'(x) E IR
9 : X --t IR , x f----+
o if g' (x) tf. IR .
Then 9 is as required (Propositions 2.8.3 e), 2.1.21 c)).
c) Take J E IRx. If f*x Jdp = -00, then the function 9 .- -ooex
satisfies the requirements. Assume therefore that
Define
1 Jdp> -00.
*x (5)
By (5), 1i is nonem pty. Choose an increasing sequence (h n ) nEIN from 1i such
that hn :::; J for every nand
sup l hndp = 1 Jdp. nElN *x *x
(Note that fLY hndf.1 = fLJhn) , by Proposition 2.3.18 b)). Assume, without
loss of generality, that f*x hndf.1 is real for every n. Then each hn must be
long to £1(X, f.1) (Proposition 2.3.11 b)). According to b), there is a sequence
= sup inf(f(xk) /\ m) = lim inf(f /\ m)(xn). nEIN k;:>:n n ..... oo
Thus f /\ m is lower semicontinuous. By what we have already proved, f /\ m E K(A)t. Now Proposition 2.3.2 f) implies that
f = V (f /\ m) E K(A)t. mEIN
b) In view of Proposition 17 and Proposition 2.3.4, b) follows from a). 0
Corollary 2.10.25 Let 9 be an increasing real-valued function on A and
suppose that f belongs to the Riesz lattice £(eg). Then for each c > 0, there
is in £(£g) a function f', upper semicontinuous on A, and a function f", lower semicontinuous on A, such that
f' ::; f ::; f" and £g(f") - £g(f') < c.
The functions f' and f" can be chosen so that for every Q E 1R, Q > 0, the
sets {I';::: Q}, {I" ::; -Q} are bounded in A. 0
The following result applies in particular to the Stieltjes functionals £g.
Theorem 2.10.26 For every positive, linear, nullcontinuous functional i on
K(A) , the following assertions hold.
a) (A, £(£), £) = (A, £1(£), Ii), and £1(£) is a Stone lattice.
b) The positive measure space (A, !J't(£1(i)), J,Jl) induced by (A, £1(£), It) is
closed, and (A, !J't(£1(£))) is a a-finite measurable space.
2.10 Stieltjes Functionals and Stieltjes Measures. Lebesgue Measure 431
Proof. a) The Stone lattice K(A) is clearly a-finite. Thus a) follows from
Corollary 2.4.14 and Proposition 2.5.18.
b) is a consequence of a), Theorem 2.6.20 and Proposition 2.6.19. 0
Definition 2.10.27 Let 9 be an increasing real-valued function on A. Then
the integral for the Daniell space (A, K(A), £g), which we denote as usual by
(A, £1(£g), It ), is called the Stieltjes integral associated with g. The positive o
measure induced on !Jt(£I(£g)) by flo shall be denoted by J1-g. The closed posi-
tive measure space (A, !Jt(£1 (£g)), J1-g) induced by the Stieltjes integral associated
with 9 is called the Stieltjes measure space associated with g. The measure
J1-g is called the Stieltjes measure on A associated with g. For f E £1 (J1-g) , we write
lfdg:= If(x)dg(x) := lfdJ1-g.
Quite generally, a positive measure J1- on a ring of subsets of A is said to be
a Stieltjes measure on A iff J1- = J1-g for some increasing real-valued function
9 on A. 0
According to Theorem 2.7.18, the Daniell-space integral called the Stieltjes
integral associated with 9 is the same as the measure-space integral of the
Stieltjes measure space associated with g; that is,
In preparation for describing what kind of sets belong to £(A, J1-g) we define
Borel sets, Ka-sets, and Go-sets, and examine a few of their properties.
Definition 2.10.28 ~(A) denotes the 8-ring generated by the subsets of A
that are closed in A. Sets belonging to ~(A) are called the Borel sets in A.
We use "Borel measurable" as a synonym for ''I.B(A)-measurable'' {provided
A is evident from the context}.
~b(A) denotes the set consisting of all Borel sets in A that are also bounded
in A.
A subset C of A is called a Ka-set in A {Fa-set in A} iff it can be
written as the union of a countable family of sets that are compact {closed}
in A. A subset C of A is called a Grset in A iff it can be written as the
intersection of a countable family of sets that are open in A. 0
432 2. Elementary Integration Theory
The reader can readily verify the following proposition.
Proposition 2.10.29 a) Intervals from A are Krr-sets in A.
b) The Krr-sets in A, the Frr-sets in A and the Grsets in A are all Borel
sets in A. 0
Proposition 2.10.30 a) The Borel sets in A form a a-algebra on A. A
subset of A is Borel measurable iff it is a Borel set.
b) ~(A) is the a-algebra on A generated by the compact subsets of A.
c) ~(A) is the a-algebra on A generated by the intervals that are closed
and bounded in A.
d) ~(A) is the a-algebra on A generated by
J := {[x, yll x, yEA, x ::; y} .
e) The Borel sets in A that are bounded in A form a 6-ring. In fact, ~b(A)
is the 6 -ring generated by the intervals that are closed and bounded in A.
f) For Be A, B is Borel-measurable iff B is ~b(A)-measurable.
Proof. a) Since A is closed in A, a) follows from Propositions 2.7.15 and
2.8.2.
b) Let <5 denote the a-algebra on A generated by the sets that are com
pact in A. Since compact sets of A are closed in A, <5 c ~(A), by a). In
view of Propositions 3 and 29 a), every set that is open in A is a Krr-set in A
and therefore belongs to <5. Hence every set that is closed in A belongs to <5, and we conclude that ~(A) C <5.
e) Denote by 9l the 6-ring generated by the intervals that are closed and
bounded in A. Since the set 'I' of all subsets of A that are bounded in A is
obviously a 6-ring, we obtain
9l c ~(A) n'I' = ~b(A) .
For D an interval that is closed and bounded in A, put
<5(D) := {B E ~(A) IBn DE 9l}.
Evidently, <5(D) is a a-algebra on A containing all intervals that are closed in
A. Since every set which is open in A can be written as the union of countably
2.10 Stieltjes Functionals and Stieitjes Measures. Lebesgue Measure 433
many closed intervals, 6(D) contains all sets that are open in A and hence
all sets that are closed in A. Thus !B(A) C 6(D) .
Now take B E !Bb(A). Choose an interval D that is closed and bounded
in A and contains B. Then B E 6(D) and hence B = B n D E 9t. Thus
!Bb(A) C 9t.
c) follows from a) and e) .
d) can easily be deduced from c) .
f) The inclusion IDt(A, !B(A)) c IDt(A, !Bb(A)) is evident. So take C in
IDt(A, !Bb(A). Choose an increasing sequence (An)nEIN of compact intervals
from A whose union is A. For each n, C n An E !Bb(A) C !B(A) . Hence
C = U (C nAn) E !B(A) . o nEIN
We can now characterize Ilg-integrable sets.
Theorem 2.10.31 Let Il be a Stieltjes measure on A.
a) Every Borel set in A is Il-measurable.
b) Every Borel set in A that is bounded in A is Il-integrable.
Moreover, given a subset D of A, the following are equivalent.
c) D is Il-integrable.
d) For every real number e > 0 there is an open set B of A and a compact
set C of A, both belonging to '£(A, Il) such that
C cDC Band J-t(B) - Il(C) < e.
e) There are a Grset B and a K,,-set C, both belonging to '£(A, Il) such
that
C cDc Band B\C E I)1(A, Il) .
f) D can be written as the disjoint union of all-integrable K,,-set in A
and a Il-null set in A.
434 2. Elementary Integration Theory
Proof a) Let C be an interval that is closed and bounded in A. Construct
f E K(A)+ such that C = {f = I}. In view of Propositions 2.9.2 d) and 2.8.3
f), C E oot(A, p,) . It follows that 23(A) c oot(A, p,) (Proposition 30 c)).
b) Take B E 23 b(A) . There is an interval C which is closed and bounded
in A and contains B. Taking f as in the proof of a), we have eB $; f. Then,
according to Corollary 2.9.11 a), eB E .c1(A,p,).
c) ::::} d). Let 9 be an increasing real-valued function on A such that
IJ = lJy and let c > 0 be given. By virtue of Theorem 26 a) and Corollary
25, there is an upper semicontinuous function f' and a lower semi continuous
function f" such that
I' $; e D $; f" ,
V Q: E R, Q: > 0, {I' ~ Q:} is bounded in A,
£g(l") - £g(l') < c/2.
The sequence (e{!'2: 1/ n })nEIN is an increasing IJ-sequence whose supremum lies between f' and e D. Therefore
and there is an m E IN such that
Put
B := {I" > I}, C:= {f' ~ ~} .
By Proposition 19, B is open in A and C is closed in A. Since C is also
bounded in A, it is compact in A (Theorem 6 a) ::::} b)). Thus, using b) ,
C E 'c(A, IJ) . By a), B is IJ-measurable, and eB $; f" . According to Corollary
2.9.11 a), BE 'c(A, IJ) . Moreover, C CDC Band
d) ::::} e) ::::} f) ::::} c). These implications are evident. o
2.10 Stieltjes Functionals and Stieltjes Measures. Lebesgue Measure 435
Corollary 2.10.32 Let I-' be a Stieltjes measure on A. Then, for every set
DE 'c(A, 1-'),
I-'(D) = inf{I-'(B) I B E 'c(A, 1-') , DeB, B is open in A}
= sup{I-'(C) ICc D, C is compact in A}.
o
Corollary 2.10.33 Every Lower semicontinuous extended-reaL-valued func
tion on A is BoreL measurabLe, ~b(A) -measurabLe, and l-'-measurabLe for every
Stieltjes measure I-' on A. The same assertions hoLd for every upper semicon
tinuous extended-reaL-vaLued function on A.
Proof Note that for each real a, the set {J < a} is open in A if f is upper
semicontinuous and is an FI1 -set in A if f is lower semicontinuous (Proposition
19), and apply Propositions 29, 30 f) and Theorem 31 a) . 0
Proposition 2.10.34 Let I-' be a Stieltjes measure on A. Let 9l be a set
ring contained in 'c(A,I-') such that 916 contains every intervaL from A that is
bounded in A. Set
v:= I-'I!R
and denote by (A, c" €) the Daniell space associated with the positive measure
space (A, 9l, v). Then
(4)
(5)
Proof By Proposition 2.5.8, every set from 916 is contained in a set from 9l.
Using the hypothesis on intervals from A, we conclude that the pair (A,9l)
is a-finite. The closure and the completion of (A, 9l, v) are therefore identical
(Proposition 2.7.12). Since (A,'c(A,I-'),I-') is a closed positive measure space
extending (A, 9l, v) , it follows that
(Proposition 2.7.2). To complete the proof of (4), we need only show that
'c(A,I-') c 'c(A, v) . Define
436 2. Elementary Integration Theory
(5 := {B E .c(A, J-L) I B E .c(A, v) and vA(B) = J-L(B) } .
Take B E .c(A, J-L) , and suppose that B is open in A. Then B can be
written as a countable disjoint union of intervals open in A. Each interval
open in A can be written as a countable disjoint union of intervals (half
open, half-closed) that are bounded in A. These latter intervals belong to (5.
Countable additivity (Theorem 2.6.4) together with Corollary 2.6.21 c) imply
that B E (5. In view of Corollary 32, every J-L-integrable Go-set in A is the
intersection of a decreasing sequence of J-L-integrable sets that are open in A.
Since .c(A, J-L)n.c(A, v) is a a-ring and J-L and vA are nullcontinuous, it follows
that J-L-integrable Go-sets in A must all belong to (5.
Now let B be a compact set of A. Choose an interval C containing B such that C is bounded and open in A. Then C belongs to .c(A, J-L) (Theorem
31 b)), C\B is open in A, and both C and C\B must belong to (5. Writing
B as C\(C\B), we see that B belongs to (5. Using Corollary 2.6.21 a) , we
conclude that J-L-integrable K,,-sets in A belong to (5.
Take B E .c(A, J-L). By Theorem 31 c) => e), there are a J-L-integrable
Go-set D and a J-L-integrable K,,-set C such that C c BcD and D\C is J-L-null. Both C and D belong to (5, so D\C is v-null, so B\C is vnull. As the union of the v-integrable set C and the v-null set B\C, B is
v-integrable. Hence .c(A, J-L) c .c(A, v), proving (4).
Finally, (5) is a consequence of (4), Corollary 2.7.9 and Proposition 2.7.12.
o
Proposition 34 is very useful. It enables us to specify simple set systems for
which the accompanying collection of measures is in one-to-one correspondence
with the set of Stieltjes measures on A. This possibility, which we pursue more
thoroughly in Chapter 6, after real Stieltjes measures have been defined, makes
the rather complicated Stieltjes measures easier to manage.
In preparation for stating the values assigned by Stieltjes measures to the
intervals from A, we make a straightforward observation, which the reader can
readily verify.
Proposition 2.10.35 Let g be an increasing real-valued function on A.
a) For every x E A\{a}, g(x-) exists and
g(x-) = sup{g(y) lyE la, x[} .
b) For every x E A\{b}, g(x+) exists and
2.10 Stieltjes Functionals and Stieltjes Measures. Lebesgue Measure 437
g(x+) = inf{g(y) lyE lx, b[} . o
Theorem 2.10.36 Let 9 be an increasing real-valued function on A. If
x, yEA and x :::; y, then
{Lg([x, y]) = g(y+) - g(x-) .
If x, yEA and x < y, then
{Lg( lx, y]) = g(y+) - g(x+) ,
{Lg( [x, y[) = g(y-) - g(x-) ,
{Lg(]x, y[) = g(y-) - g(x+) .
For each x E A, {x} is a {Lg -null set iff 9 is continuous at x.
Proof Let x and Y belong to A with x :::; y. First, choose from A a sequence
(Xn)nEIN that is either strictly increasing (if x =1= a) or constant (if x = a) but
in any case has its supremum equal to x. Next, choose from A a sequence
(Yn)nEIN that is either strictly decreasing (if Y =1= b) or constant (if y = b) but
in any case has its infimum equal to y. For n in 1N, define
1 if Z E [x, y]
0 if Z E A\[xn, Yn]
fn: A --+ JR, Z t---+ Z - Xn X-Xn
if Z E [xn,x[
Yn - Z if Z E ]y, Yn] .
Yn - Y
Note that Un)nEIN is a decreasing sequence from K(A) whose infimum is e[x,y] •
Hence
(6)
Consider the partition
of the interval [xn, Yn]. It is easy to calculate CP* Un, g; Pn) and cp* Un' g; Pn) , and to verify that
438 2. Elementary Integration Theory
Since
for every n E IN, it follows from (6) that
J-tg([x, yl) = g(y+) - g(x-) .
Now suppose that x, yEA and x < y. Since
[x, yl = lx, yl U [x, xl = [x, y[ U [y, y],
[x, y[ = lx, y[ u [x, xl
and these are all disjoint unions, the remaining formulas follow from the formula
already established.
From what has already been proved, J-tg({x}) = g(x+) - g(x-) for every
x in A. Thus J-tg ( {x}) = 0 iff g is continuous at x. 0
Corollary 2.10.37 Suppose that A = la, b[. Take x E A and let g be an increasing real function on A.
a) la, x[ E £(J-tg) - la, xl E £(J-tg) - g(a+) f= -00, and in this case
J-tg( la, x[) = g(x-) - g(a+) ,
J-tg( la, xl) = g(x+) - g(a+) .
b) lx, b[ E £(J-tg) - [x, b[ E £(J-tg) - g(b-) f= 00, and in this case
J-tg(lx,b[) = g(b-) - g(x+),
J-tg([x, b[) = g(b-) - g(x-) .
c) A E £(J-tg) - g is bounded - J-tg is bounded, and in this case
J-tg(A) = g(b-) - g(a+) .
d) J-tg = 0 - g is constant. o
2.10 Stieltjes Functionals and Stieltjes Measures. Lebesgue Measure 439
Corollary 2.10.38 Let g be an increasing real-valued function on A. Then
the following are equivalent.
a) The function g is continuous on A.
b) Every countable subset of A is J.Lg-null.
c) For every x E A, the set {x} is J.Lg -null. o
We now come to the most important example of a StieJtjes measure, namely
Lebesgue measure. This measure distinguishes itself from other StieJtjes mea
sures by its translation invariance, its special link with the algebraic structure
of IR.
Definition 2.10.39 The Stieltjes measure J.Lg associated with the identity
function
g:A---+A, Xl---+X
is called the Lebesgue measure on A . We denote the Lebesgue measure on A
by AA. The integral associated with AA is called the Lebesgue integral on A.
We use "Lebesgue measurable on or in A", "Lebesgue integrable on or in
A" and "Lebesgue null" as synonyms for "AA-measurable", AA-integrable",
and "AA -null", respectively. -A
For f E IR and x, yEA, if x ~ y and fejx,y[ E £1 (AA), we define
l Y f(t)dt:= lY
fdt:= ! fejx,y[dAA,
luX f(t)dt:= luX fdt := -lY f(t)dt. 0
Corollary 2.10.40 Every countable subset of A is Lebesgue null. The Lebesgue
measure on A is bounded iff the interval A is bounded in IR.
Proof Apply Corollaries 37 c) and 38. o
Theorem 2.10.41 (Translation invariance 0/ Lebesgue measure on IR).
Let A be Lebesgue measure on IR. Let f be an extended-real-valued function
on IR and B a subset of IR. For "I E IR, define
R, := {x + "I I x E B},
Iy : IR ---+ IR, X 1---+ f(x - "I) .
Let "I be a real number.
440 2. Elementary Integration Theory
a) I is Lebesgue integrable on JR iff 1'"( is Lebesgue integrable on JR, and
in this case
b) B is Lebesgue integrable in JR iff B'"( is Lebesgue integrable in JR, and
in this case
c) I is Lebesgue measurable on JR iff 1'"( is.
d) B is Lebesgue measurable in JR iff B'"( is.
Prool. a) Let J be the set of all bounded intervals from JR. It is easily verified that
6 := {U A. I (A.).EI is a finite family from J} .EI
is a ring of sets. Hence 6 is the set-ring generated by J. Put v:= Ais . Then, by Proposition 34,
(7)
Fix 'Y E JR and put
In view of (7), it suffices to prove that :F = £1(JR, v). We use the Induction
Principle.
By the formulas of Theorem 36, every 6-step function belongs to :F. The
reader can readily verify that lim In E:F for every v-sequence (fn)nEIN from n-+oo
:F. Finally, take I E :F and g E JRIR with g = I v-a.e. If we can show that
B'"( E IJt(JR, v) whenever B E IJt(JR, v) , it will follow that g E :F. But in view
of Proposition 2.7.12 b), this is a consequence of the characterization of null
sets given in Theorem 2.7.6 k) and the formulas from Theorem 36.
It follows from Theorem 2.4.17 that :F = £1 (JR, v) , as claimed.
b) Note that (eB)'"( = e(B~) , and apply a) .
c) Suppose IE M(JR, A)+ and 'Y E JR. By b), C(_'"() belongs to ..c(JR, A) for every Lebesgue integrable set C. Applying Proposition 2.9.13 a) => c) to
2.10 Stieltjes Functionals and Stieltjes Measures. Lebesgue Measure 441
the measure space (JR, £(JR, A), A), we conclude that f /\ aec(_,) belongs to
C} (JR, A) for every positive real number a and every Lebesgue integrable set
C. By a), f-y /\ aec belongs to C} (JR, A) for every positive real number a and
every Lebesgue integrable set C. Now Proposition 2.9.13 c) ~ a) shows that
f,y is Lebesgue measurable.
For arbitrary Lebesgue measurable f on JR, note that f = f+ - f- , and
apply what has already been proved to the functions f+ , f- .
d) This assertion follows from c) . o
An analogous theorem obviously holds for Lebesgue measure on an arbitrary
interval A, provided we restrict "I so that AA(B-y) and f f-ydA A are defined.
An interesting question arises: Is every subset of JR Lebesgue measurable?
The following non-trivial example, answering this question, is due to Vitali
(1905).
Theorem 2.10.42 Every Lebesgue integrable set in JR, whose Lebesgue mea
sure is strictly positive, has a subset that is not Lebesgue measurable in JR.
Proof. Take B E £(lR, AIR) with AIR(B) > O. Without loss of generality
assume, for some m in IN, that
Be [-m,m].
Define an equivalence relation on B as follows:
x - y:- x - y E (Q.
Choose one element (Axiom of Choice!) from each equivalence class, and let C
be the set consisting of the chosen elements of B. We claim that the subset C
is not AIR-measurable.
Let
<p: IN ----+ [-2m, 2m] n(Q
be bijective. For n in IN, define
Cn := {x + <p(n) I x E C}.
Then (Cn)nEIN is a disjoint sequence, and
Be U Cn e [-3m, 3m]. (8) nEIN
442 2. Elementary Integration Theory
As a subset of a Ant-integrable set, if C were Ant-measurable, it would also
be Ant-integrable. Since Ant is translation invariant, each Cn would also be
Ant-integrable and satisfy
In view of (8), we would have
0< Ant(B) ::; L:* Ant(Cn ) ::; 6m neIN
hence
for every n in IN. This last inequality is not possible. Thus C cannot belong
to VJ1(JR, Ant) . 0
Exercises
E 2.10.1 (E)
We offer a different way to construct Stieltjes measures on A = la, b[ . Let
9 : A -+ JR be increasing and left-continuous, and define
Vg : 3 ---t JR, [x, yl f---t g(y) - g(x) .
(3 denotes the set of all right half-open intervals in A.)
Prove the following propositions.
(a) If B E 3 and if (A.),El is a finite disjoint family from 3 with U A, c B, ,el
then 0 ::; E vg(A,) ::; vg(B) . ,el
({J) If BE 3 and if (A.),El is a finite disjoint family from 3 with B = U A. , ,el
then vg(B) = E vg(A,) . ,el
(-y) If B E 3 and if (A.),el is a finite family from 3 with B c U A., then ,el
vg(B) ::; E vg(A.) . ,el
(6) vg(B) 2:: 0 for any B E 3.
(c) Vg is a u-additive positive content on 3.
2.10 Stieltjes Functionals and Stieltjes Measures. Lebesgue Measure 443
(Take [x, y[ E .j and let ([xn' Yn[ )nEIN be a disjoint sequence from .j with
[x, Y[ = U [xn' Yn[ . Take E: > O. By the left-continuity of g, there are a Z E nEIN
lx, y[ and Zn E A, Zn < Xn (for each n E IN) with g(y) - g(z) < E:/2 and
g(xn) - g(zn) < E:/2n +l for each n E IN. [x, zl c U lZn' Yn[ , and so there is a nEIN
finite N c IN with [x, z[ c U ]Zn' Yn[ . Using ('y), g(y) - g(x):s L (g(Yn)-nEN nEIN
g(xn )) + E: . Thus
g(y) - g(x) :S :L)g(Yn) - g(xn)) nEIN
since E: was arbitrary. The converse follows from (a), showing the a-additivity
of IIg .)
(() If 9 E lRA is an increasing left-continuous function, then there is exactly
one positive measure, Jig, on the ring of sets of the interval forms of
A, .jr, with Jig ([x, y[) = g(y) - g(x) for each [x, y[ E .j with x :S y.
(7]) Given a positive measure on .jr, /1, there is an increasing left-continuous
function 9 E lRA with /1 = Jig.
(19) If 9 and h are increasing left-continuous functions on A with Jig = Jih ,
then 9 = h + I for some I E lR.
Of course, we have Jig = /1glJr , where /1g denotes the Stieltjes measure associ
ated with g.
The reader may wonder why we have carried out the constructions of this
exercise only for left continuous g. The next exercise shows that this is no real
restriction.
E 2.10.2 (E)
Prove the following.
(a) Let 9 E lRA be increasing. For each x E A, define ax := g(x+) - g(x-) .
Then ax 2: 0 for each x E A, and 9 is continuous at x iff ax = 0 .
((3) If [a, (3l c A, then (ax)xE[et,lll is summable, and L ax:S g((3) - g(a) . xE [et,lll
('y) The set of points at which 9 is discontinuous is countable.
(0) If gl and g2 are increasing functions on A with the same points of
continuity such that gl(X) = g2(X) in each point x of continuity, then
/1g1 = /1g2 .
444 2. Elementary Integration Theory
(c) If g is an increasing function on A, then there is exactly one left
continuous increasing function 9 on A such that g(x) = g(x) in every
point x of continuity of g.
() J-Lg restricted to the ring of sets of the interval forms of A is identical with
jig in the sense of Exercise 2.10.1.
(77) If g is an increasing function on A, then
E 2.10.3 (C)
Generalized Cantor sets. Let (On)nEIN be a sequence of numbers in 10, 1 [. Denote
by ).,(I) the length of the interval I em.. Take a, b Em., a < b, and set
,:=(a+b)/2. From the interval [a, bl remove an open interval, Ill, of length odb - al
centered on ,. This leaves two disjoint closed intervals, J u and J 12 . Now
remove from J li the open interval I2i of length (2).,(Jli) centered on the
midpoint of J li (i = 1,2). This yields four pairwise disjoint closed intervals,
J 21 , J 22 , J 23 , J 24 . Next remove the open intervals I3i of length (3).,(J2i ) cen
tered on the midpoint of J2i . This leaves 8 (= 23) pairwise disjoint closed
intervals J3i (i E {I, 2, ... , 8}). Continuing in this way, construct, for each
n E IN, 2n pairwise disjoint closed intervals Jni (i E IN 2n ) . The set
is called the generalized Cantor set determined by (On)nEIN. The set
is the classical Cantor set. (H.I.S. Smith, 1875; Volterra, 1881; Cantor, 1883.)
Denote by Ct (resp. CT ) the set of left (resp. right) endpoints of the inter
vals Jni (n E IN, i E IN2n). Prove the statements that follow.
(0) Ct and CT are countably infinite disjoint subsets of C .
(/3) Given xEC and c>O, lX-c,xlnct #0 and [x,x+c[nCT #0.
b) Every point of C is a point of accumulation both of Ct and CT'
2.10 Stieltjes Functionals and Stieltjes Measures. Lebesgue Measure 445
(0) C is a perfect set, i.e. C coincides with the set of its points of accumu
lation.
(c) C has no interior points.
(() C is nowhere dense in JR.
(1]) There is a bijection from C onto the set of all sequences from {O, I}.
Hence C has the same cardinality as JR.
From now on, denote by C the classical Cantor set.
(19) There is exactly one increasing function 9 on [0,1] such that 9 is equal
to (2i - I)/2n on Ini for each n E IN and i E IN2n-l • We call 9 the
Cantor function.
(L) The Cantor function 9 is continuous, and g(O) = 0, g(I) = 1.
(II:) J1-g(C) = 1, but C is a Lebesgue null set.
(..\) Take a E [0,1] and define h(x) := ag(x) + (1- a)x for every x E [0,1].
Then h is an increasing function on [0,1] such that J1-h(C) = a and
h(O) = 0, h(I) = 1.
(J1-) Define
{ 0 for x E C
f : [0,1] --t JR, x ~ n for x E Ini.
Then J f d..\ = 3, where ..\ is the Lebesgue measure on [0, 1], and
J fdJ1-g = 0 where 9 is the Cantor function.
(v) Set 9t:= {[a,,8[ \C I a,,8 E [0, I]} and
v: 9t --t JR, [a,,8[\C ~ g(,8) - g(a).
Then the following are true.
(VI) 9t is a semi-ring.
(V2) v is an additive positive content which is nullcontinuous.
(V3) v is not a-additive.
(0 If J1- is a positive additive real function on the semi-ring {[x, y[ I x, Y E
JR}, then J1- is a-additive iff it is nullcontinuous.
