An introduction to Geometric Measure Theory Part...

An introduction to Geometric Measure TheoryPart 2: Hausdorff measure

Toby O’Neil, 10 October 2016

TCON (Open University) An introduction to GMT, part 2 10 October 2016 1 / 40

Last week. . .

• Discussed several motivating examples• Defined box dimension• Introduced Hausdorff measure and dimension


Today

1 Determine the Hausdorff dimension of a ‘simple’ set2 Look at properties of Hausdorff measure. (Prove that it’s a

measure. . . )3 Find some useful alternative characterisations of Hausdorff

dimension4 Discuss an application


Recall: covers

Definition (r -covers)Let r > 0. A countable collection of sets {Ui : i 2 N} in Rn is anr -cover of E ✓ Rn if

1 E ✓ Si2N Ui

2 diam(Ui) r for each i 2 N.


notation

For a set A and s � 0, we define

|A|s =

8><

>:

0 if A= ;,1 if A 6= ; and s = 0,diam(A)s otherwise.


Hausdorff measures

Definition (s-dimensional Hausdorff measure)Suppose that F is a subset of Rn and s � 0. For any r > 0, wedefine

Hsr (F ) = inf

( 1X

i=1

|Ui |s : {Ui} is an r -cover of F

).

The s-dimensional Hausdorff measure is then given by

Hs(F ) = limr&0

Hsr (F ).


Easy properties of Hausdorff measure

1 If s < t and Hs(F ) < 1, then Ht(F ) = 0. (So for t > n,Ht(Rn) = 0.)

2 If s is a non-negative integer, then Hs is a constant multipleof the usual s-dimensional volume. (counting measure,length measure, area, volume etc.) — we shall not provethis.


More easy properties of Hausdorff measure

1 If 0 < r < inf{d(x , y) : x 2 E , y 2 F}, then

Hsr (E [ F ) = Hs

r (E) +Hsr (F )

and so, if inf{d(x , y) : x 2 E , y 2 F} > 0, then

Hs(E [ F ) = Hs(E) +Hs(F ).

2 Let F ✓ Rn and suppose that f : F ! Rm is such that forsome fixed constants c and ↵

|f (x)� f (y)| c|x � y |↵ whenever x , y 2 F .

Then for each s, Hs/↵(f (F )) cs/↵Hs(F ).


Hausdorff dimension

Definition (Hausdorff dimension)For a set F , we define the Hausdorff dimension of F by

dimH(F ) = inf{s � 0 : Hs(F ) = 0} = sup{s : Hs(F ) = 1}.

Observations1 Hausdorff dimension is defined for any set F (unlike box

dimension).2 dimH(;) = 03 If A is a countable set, then dimH(A) = 0. In particular, Q

has Hausdorff dimension 0.


Properties of Hausdorff dimension

1 If A ✓ B, then dimH(A) dimH(B).2 If F1,F2,F3, . . . is a (countable) sequence of sets, then

dimH

1[

i=1

Fi

!= sup{dimH(Fi) : 1 i 1}.

3 for bounded sets F , dimH(F ) dimB(F ).


Calculating Hausdorff dimension

(14 ⇥ 1

4)-Cantor set


Heuristic for finding what the dimension could beAssume that F has positive and finite s-dimensional Hausdorffmeasure when s = dimH(F ) and then represent F as a finitedisjoint union of scaled copies of F , Fi , say where Fi is a copy ofF scaled by �i . Then

Hs(F ) = Hs

[

i

Fi

!=X

i

Hs(Fi) =X

i

�si Hs(F ).

Dividing through by Hs(F ) then gives

1 =X

i

�si .

For (14 ⇥ 1

4)-Cantor set obtain 1.


Covers provide an upper bound

If set is bounded, then can use box dimension to find an upperbound, since in this case

dimH(F ) dimB(F ).


Upper bound for our example.



Lower bound for our example



Digression: some measure theory(X , d), a metric space. (Usually complete and separable)

Definition: (outer) measuresA set function µ : {A : A ✓ X} ! [0,1] is a measure if

1 µ(;) = 02 if A ✓ B, then µ(A) µ(B)3 µ (

S1i=1 Ai)

P1i=1 µ(Ai)

Definition: measurable setA ✓ X is measurable if

µ(E) = µ(E \ A) + µ(E \ A) for each E ✓ X


Basic results

TheoremIf µ is a measure and M is the µ-measurable sets, then

1 M is a �-algebra. [; 2 M, closed under complements andcountable unions]

2 if µ(A) = 0, then A 2 M3 if A1,A2, . . . 2 M are pairwise disjoint, then

µ (S

Ai) =P1

i=1 µ(Ai)4 if A1,A2, . . . 2 M, then

1 µ (S

Ai) = limi!1 µ(Ai), provided A1 ✓ A2 ✓ · · ·2 µ (

TAi) = limi!1 µ(Ai), provided A1 ◆ A2 ◆ · · · and

µ(A1) < 1


Regular measuresµ is a regular measure if for each A ✓ X , there is aµ-measurable set B with A ✓ B and µ(A) = µ(B).

LemmaSuppose that µ is a regular measure.If A1 ✓ A2 ✓ · · · , then

µ

1[

i=1

Ai

!= lim

i!1µ(Ai).


Identifying measurable sets

DefinitionThe family of Borel sets in a metric space X is the smallest�-algebra that contains the open subsets of X .

DefinitionA measure µ is:

1 a Borel measure if the Borel sets are µ-measurable2 Borel regular if it is a Borel measure and for each A ✓ X ,

there is a Borel set B with A ✓ B and µ(A) = µ(B).


TheoremLet µ be a measure on X. Then µ is a Borel measure if, andonly if,

µ(A [ B) = µ(A) + µ(B),

whenever inf{d(x , y) : x 2 A, y 2 B} > 0.

TheoremHs is a Borel regular measure for each s � 0.


Definition (support)The support of a Borel measure µ is defined to be the smallestclosed set F for which µ(X \ F ) = 0.spt(µ) = X \S{U : U is open and µ(U) = 0}.

Definition (Mass distribution)A Borel measure µ is a mass distribution on the set F if thesupport of µ is contained in F and 0 < µ(F ) < 1.


Mass distribution principleSuppose that µ is a mass distribution on a set F and for somereal number s, there are c > 0 and r0 > 0 so that

µ(U) c|U|s if diam(U) r0.

Then Hs(F ) � µ(F )/c > 0 and so dimH(F ) � s.


Cantor set revisited



Approximating sets

TheoremLet µ be a Borel regular measure on X, let A be a µ-measurableset and let ✏ > 0.

1 If µ(A) < 1, then there is a closed set C ✓ A for whichµ(A \ C) < ✏.

2 If there are open sets V1,V2, . . . with A ✓ S1i=1 and

µ(Vi) < 1 for each i, then there is an open set V withA ✓ V and µ(V \ A) < ✏.


Radon measuresDefinitionA Borel measure µ is a Radon measure on X if

1 all compact subsets of X have finite µ-measure2 for open sets V ,

µ(V ) = sup{µ(K ) : K ✓ V is compact}3 for each set A ✓ X ,

µ(A) = inf{µ(V ) : V is open and A ✓ V}.

CorollaryA measure µ on Rn is a Radon measure if and only if it is locallyfinite and Borel regular.


Energies

DefinitionFor A ✓ Rn, let M(A) denote the set of all compactly supportedRadon measures µ with 0 < µ(A) < 1 and with supportcontained in A.

Definition (t-energy)For a Radon measure µ on Rn and t � 0, we define the t-energyof µ by

It(µ) =ZZ

1|x � y |t dµ(x)dµ(y).

(This may be infinite.)


Aside: integration

Theorem (Fubini)Let X and Y be separable metric spaces with µ and ⌫ locallyfinite Borel measures on X and Y , respectively.If f is a non-negative Borel function on X ⇥ Y, then

ZZf (x , y) dµ(x)d⌫(y) =

ZZf (x , y) d⌫(y)dµ(x).

In particular, if f is the characteristic function of a Borel set, thenZ

µ({x : (x , y) 2 A}) d⌫(y) =Z

⌫({y : (x , y) 2 A}) dµ(x).


More integration: a useful equation

TheoremLet µ be a Borel measure and f a non-negative Borel functionon a separable metric space X. Then

Zf dµ =

Z 1

0µ({x 2 X : f (x) � t}) dt .


Another view of It(µ)

It(µ) = tZ 1

0r�t�1µ(B(x , r)) dr


A useful characterisation of dimH

TheoremIf A is a Borel set in Rn, then

dimH(A) = sup{t : There is µ 2 A with It(µ) < 1}.(Finer results are available. . . )


Date post:	13-Jul-2020
Category:	Documents
Upload:	others
View:	3 times
Download:	0 times

An introduction to Geometric Measure Theory Part...

Documents