Continuity properties of Markov semigroups and their restrictions to invariant L 1 -spaces Sander C. Hille, Dani¨ el T. H. Worm == Report MI-2009-04 == Mathematical Institute, University Leiden P.O. Box 9512, 2300 RA Leiden, The Netherlands E-mail: {shille,dworm}@math.leidenuniv.nl March 20, 2009 Abstract We consider Markov semigroups on the cone of positive finite measures on a complete separable metric space. Such a semigroup extends to a semigroup of linear operators on the vector space of measures that typi- cally fails to be strongly continuous for the total variation norm. First we characterise when the restriction of a Markov semigroup to an invariant L 1 -space is strongly continuous. Aided by this result we provide several characterisations of the subspace of strong continuity for the total vari- ation norm. We prove that this subspace is a projection band in the Banach lattice of finite measures, and consequently obtain a direct sum decomposition. 1 Introduction Markov operators on the cone of positive finite measures are additive and pos- itively homogeneous operators on this cone that preserve mass, i.e. the total variation norm of measures. A Markov semigroup is a semigroup of Markov op- erators. They naturally occur in probability theory and the theory of Markov processes [12, 18]. Moreover, one encounters such semigroups also in the set- ting of measure-valued structured population models (cf. e.g. [6, 7] and an application to cell growth and division in [5]). Here the measure represents the constitution of the population at each time. The Markov semigroups that are obtained in both settings are hardly ever con- tinuous for the total variation norm k·k TV on the space of finite measures M(S) on the underlying measurable space (S, Σ), typically a complete separable metric space with its Borel σ-algebra. Notable exceptions are Markov jump processes [12, 13], which yield strongly continuous semigroups in M(S) for k·k TV when (S, Σ) is merely a measurable space as above [29]. This may have motivated 1
Transcript
Continuity properties of Markov semigroups and
their restrictions to invariant L1-spaces
Sander C. Hille, Daniel T. H. Worm
== Report MI-2009-04 ==
Mathematical Institute, University LeidenP.O. Box 9512, 2300 RA Leiden, The Netherlands
We consider Markov semigroups on the cone of positive finite measureson a complete separable metric space. Such a semigroup extends to asemigroup of linear operators on the vector space of measures that typi-cally fails to be strongly continuous for the total variation norm. First wecharacterise when the restriction of a Markov semigroup to an invariantL1-space is strongly continuous. Aided by this result we provide severalcharacterisations of the subspace of strong continuity for the total vari-ation norm. We prove that this subspace is a projection band in theBanach lattice of finite measures, and consequently obtain a direct sumdecomposition.
1 Introduction
Markov operators on the cone of positive finite measures are additive and pos-itively homogeneous operators on this cone that preserve mass, i.e. the totalvariation norm of measures. A Markov semigroup is a semigroup of Markov op-erators. They naturally occur in probability theory and the theory of Markovprocesses [12, 18]. Moreover, one encounters such semigroups also in the set-ting of measure-valued structured population models (cf. e.g. [6, 7] and anapplication to cell growth and division in [5]). Here the measure represents theconstitution of the population at each time.
The Markov semigroups that are obtained in both settings are hardly ever con-tinuous for the total variation norm ‖·‖TV on the space of finite measuresM(S)on the underlying measurable space (S,Σ), typically a complete separable metricspace with its Borel σ-algebra. Notable exceptions are Markov jump processes[12, 13], which yield strongly continuous semigroups in M(S) for ‖ · ‖TV when(S,Σ) is merely a measurable space as above [29]. This may have motivated
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other researchers to consider the more restrictive setting of strongly continuousMarkov semigroups on L1-spaces with respect to particular positive measures(see e.g. [19, 27, 28]). In view of the above mentioned applications this settingseems to be too restrictive however.
In this paper we consider Markov semigroups (P (t))t≥0 on the positive finiteBorel measures M+(S) on a complete separable metric space (S, d). The pos-itive operators P (t) naturally extend to bounded linear operators P (t) on theBanach lattice (M(S), ‖ · ‖TV. We address two closely related questions: when(P (t))t≥0 leaves invariant a cone Γ ⊂ M+(S) such that the measures in Γ areall absolutely continuous with respect to a single measure µ, i.e. Γ = L1
+(S, µ),it induces a semigroup of nonexpansive linear operators on L1(S, µ) that areisometries on L1
+(S, µ). The first question is then to characterise when thisinduced semigroup is strongly continuous. This is achieved in Theorem 4.6,partially using an argument inspired by [16], under the assumption that foreach µ ∈ M+(S), the map t 7→ P (t)µ : R+ → M+(S) is continuous for therelative topology on M+(S) of the weak∗-topology on C∗b (S). It was shown in[9, 10] that this topology is metrisable by means of the norm on BL(S)∗, the dualof the bounded Lipschitz functions on S. See also [17] for further exploration ofthis property.
Also under this assumption and the additional assumption that each operatorP (t) : M+(S) → M+(S) is continuous for this weak∗-topology, we then char-acterise in Theorem 5.6 and Theorem 5.7 the subspace M(S)0
TV of M(S) thatconsists of all measures µ for which t 7→ P (t)µ is continuous for ‖ · ‖TV. To thatend we exploit results of [14] on modules of Banach algebras with approximateidentity and properties of Bochner integration in the Banach space SBL, whichis the closure of M(S) in BL(S)∗. These properties are of separate interest.We state and prove them in Section 2.3. A consequence of this characterisa-tion is that M(S)0
TV is dense in M(S) for the SBL-topology. In particular itis non-trivial and not ‘too small’. Moreover, it turns out to be a projectionband in the Banach lattice M(S) (Proposition 6.1), hence it is complemented.This complement is characterised and will not be (P (t))t≥0-invariant in gen-eral (unfortunately). An additional result of our approach is a generalisationof classical result by Wiener and Young ([30]) for general Markov semigroups(Theorem 6.7).
Some notational conventions. We write (Ω,Σ) to denote a measurable space,M+(Ω) to denote the cone of positive finite measures on Σ and M(Ω) the realvector space of all signed finite measures. Throughout this paper (S, d) willdenote a complete separable metric space, viewed as a measurable space withrespect to its Borel σ-algebra, with at least two elements. We write 11E forthe indicator function of E ⊂ S. 11S will be simplified to 11. For f : Ω → Rmeasurable and µ ∈M(Ω) we write 〈µ, f〉 for
∫Ωf dµ.
2 Preliminaries on spaces of measures
M(Ω) endowed with the total variation norm ‖ · ‖TV is a Banach space. Letµ, ν ∈M(Ω). µ is absolutely continuous with respect to ν, µ ν, if |µ|(E) = 0
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for every E ∈ Σ for which |ν|(E) = 0. So µ ν if and only if |µ| |ν|.
Let µ ∈M(Ω), ν ∈M+(Ω), then µ ν if and only if µ(E) = 0 for every E ∈ Σsuch that ν(E) = 0, which is easy to prove.
Since S is a complete separable metric space, it is a standard result that everyµ ∈ M(S) is inner regular, i.e. for every Borel set E in S, there are compactKn ⊂ E, such that |µ|(Kn)→ |µ|(E).
Lemma 2.1. Let µ ∈M(S), ν ∈M+(S). Then the following are equivalent:
(i) µ ν
(ii) µ(K) = 0 for all compact K in S such that ν(K) = 0.
Proof. (i) ⇒ (ii): Trivial. (ii) ⇒ (i): Let E be a Borel set in S such thatν(E) = 0. Then ν(K) = 0 for all compact K such that K ⊂ E, hence µ+(K) =µ−(K) for all compact K ⊂ E. Since µ+ and µ− are inner regular on all Borelsets, there are compact sets Kn ⊂ E, such that limn→∞ µ+(Kn) = µ+(E) andlimn→∞ µ−(Kn) = µ−(E). So µ+(E) = µ−(E) and µ(E) = µ+(E)− µ−(E) =0.
2.1 Space of measures viewed as Banach lattice
We refer to [2], [21] and [32] for the basic theory on Riesz spaces and Banachlattices.
M(Ω) is an ordered vector space for the partial ordering defined by
µ ≤ ν whenever µ(E) ≤ ν(E) for all E ∈ Σ.
M(Ω) is a Riesz space, where the least upper bound of µ and ν is given by
µ ∨ ν(E) := supµ(A) + ν(E \A) |A ∈ Σ, A ⊂ E,
and the greatest lower bound is given by
µ ∧ ν(E) := infµ(A) + ν(E \A) |A ∈ Σ, A ⊂ E.
Note that |µ| ≤ |ν| implies µ ν. The positive and negative part of µ ∈M(Ω)as introduced in measure theory, µ+ and µ−, correspond to the concepts ofpositive and negative part in a Riesz space: µ+ = µ ∨ 0, µ− = (−µ)+ and|µ| = µ+ + µ−. µ, ν ∈ M(Ω) are mutually singular, µ ⊥ ν, if there is a U ∈ Σ,such that µ(E) = µ(E ∩ U) and ν(E) = µ(E \ U) for every E ∈ Σ. Mutualsingularity of µ, ν ∈M(Ω) corresponds to the concept of disjointness in a Rieszspace: µ and ν are disjoint, µ ⊥ ν, whenever |µ| ∧ |ν| = 0. M(Ω) is a Dedekindcomplete Riesz space ([21, 1.1 Example vi)]).
M(Ω) is a Banach lattice for the total variation norm: ‖µ‖TV = |µ|(Ω), and‖ · ‖TV is an L-norm: ‖µ+ ν‖TV = ‖µ‖TV + ‖ν‖TV for all µ, ν ∈M+(Ω). As inall Banach lattices, the lattice operations are continuous for the norm topology.(see e.g. [21, Proposition 1.1.6]).
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We will now define some concepts in Riesz spaces that we will need later on: LetX be a Riesz space. A subspace I of X is an ideal of X if |x| ≤ |y| for some y ∈ Iimplies x ∈ I. An ideal B of X is a band of X if sup(A) ∈ B for every subsetA ⊂ B which has a supremum in X. A band B of X is a projection band if thereexists a bounded linear projection P : X → B, such that 0 ≤ Px ≤ x for allx ∈ X+. In this case X = B⊕B⊥, where B⊥ := x ∈ X : x ⊥ y for all y ∈ B.
In a remark in [2] (under definition 4.20) it is shown that every L-space has ordercontinuous norm as a consequence of [21, Theorem 2.4.2]. Furthermore, in aBanach lattice with order continuous norm, every closed ideal is a projectionband ([21, Corollary 2.4.2]). These statements imply
Theorem 2.2. Every closed ideal in M(Ω) is a projection band.
2.2 The space SBL
In this section we recall some definitions and results from [17]. BL(S) denotesthe Banach space of bounded real-valued Lipschitz functions for the metric d,endowed with the norm ‖f‖BL := |f |Lip + ‖f‖∞, where
|f |Lip := sup|f(x)− f(y)|
d(x, y): x, y ∈ S, x 6= y.
.
The Dirac functionals δx(f) := f(x) for x ∈ S are in BL(S)∗. We denote theusual dual norm on BL(S)∗ by ‖ · ‖∗BL.
BL(S) is in fact isometrically isomorphic to the dual of a separable Banach spaceSBL, which can be defined as the closure of the finite linear span of the δx, x ∈ S,in BL(S)∗. Then, as shown in [9, Lemma 6], each µ ∈ M(S) defines a uniqueelement in BL(S)∗, which we will also denote by µ, by sending f ∈ BL(S) to〈µ, f〉 =
∫Sf dµ. A function f ∈ BL(S) defines a bounded linear functional on
SBL by sending φ to φ(f).
By [17, Theorem 3.9 and Corollary 3.10], M+(S) is a closed convex cone ofSBL, and M(S) is a ‖ · ‖∗BL-dense subspace of SBL.
The following lemma follows from [9, Theorem 6 and Theorem 8]:
Lemma 2.3. Let µn, µ ∈ M+(S). Then ‖µn − µ‖∗BL → 0 if and only if∫Sf dµn →
∫Sf dµ for all f ∈ Cb(S).
Moreover, the restriction of the weak-star topology on Cb(S)∗ toM+(S) equalsthe restriction of the norm topology on SBL to M+(S) by [9, Theorem 18].
LetS+
BL := φ ∈ SBL : φ(f) ≥ 0 for all f ∈ BL(S), f ≥ 0.
Then S+BL =M+(S) by [17, Corollary 4.2].
When M(S) and M+(S) are equipped with the ‖ · ‖∗BL-topology, we writeM(S)BL and M+(S)BL respectively. When we use the ‖ · ‖TV-topology, wewrite M(S)TV and M+(S)TV.
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2.3 Bochner integration of SBL-valued functions
In this section we give some results on functions p : Ω→ S+BL which are strongly
measurable in the sense of Bochner. We will make use of the Monotone ClassTheorem for functions, which we state here for convenience (see e.g.[31, TheoremII.4]).
Theorem 2.4. Let E be a π-system for S and let H be a vector space of functionsfrom S to R such that
1. H contains the indicator function 11E of every E ∈ E, and H contains 11S,
2. if (fn)n is a sequence of elements of H with fn ≥ 0 and fn ↑ f , where fis bounded, then f ∈ H.
Then H contains every bounded real-valued function which is measurable withrespect to the σ-algebra generated by E.
Proposition 2.5. Let p : Ω→ S+BL. Then the following conditions are equiva-
lent:
(i) p is strongly measurable.
(ii) For each bounded measurable f : S → R, the map Ω → R : ω 7→ 〈p(ω), f〉is measurable.
(iii) For each Borel measurable E ⊂ S, the map Ω → R : ω 7→ p(ω)(E) ismeasurable from Ω to R.
Proof. (i)⇒ (ii): Let H the vector space of measurable functions h from S toR, such that ω 7→ 〈p(ω), h〉 is measurable from Ω to R. Let C be the π-systemof closed sets in S. Our aim is to show that H and C satisfy the conditions ofTheorem 2.4. Then it follows that H contains every bounded Borel measurablefunction on S.
Since p is strongly measurable, it is weakly measurable. Let C be a closedset in S and let gn(x) := max(1 − nd(x,C), 0). Then gn ∈ BL(S) ∼= S∗BL,hence Gn : ω 7→ 〈p(ω), gn〉 is measurable from Ω to R. Since C is closed,gn(x) → 11C(x) for every x ∈ S. Fix ω ∈ Ω. Then all gn are in L1(p(ω)), thus11C is in L1(p(ω)) and
limn→∞
Gn(ω) = 〈p(ω), 11C〉
by the Lebesgue Dominated Convergence Theorem. So the function
ω → p(ω)(C) = 〈p(ω), 11C〉
is the pointwise limit of measurable functions, hence measurable, which impliesthat 11C ∈ H for all closed C ⊂ S. Suppose hn ∈ H such that 0 ≤ hn ↑ h ≤M ,for some function h : Ω → R, bounded by M > 0. Then by assumptionHn : ω 7→ 〈p(ω), hn〉 is measurable for all n ∈ N. Fix ω ∈ Ω. By the LebesgueMonotone Convergence Theorem h ∈ L1(p(ω)) and limn→∞Hn(ω) = 〈p(ω), h〉.This implies that the function
ω 7→ 〈p(ω), h〉
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is the pointwise limit of measurable functions, hence measurable. So h ∈ H andthe conditions of Theorem 2.4 are satisfied.
(ii)⇒ (iii): Let E ⊂ S be measurable, then 11E is a bounded measurable func-tion from S to R.
(iii)⇒(i): By assumption ω 7→ 〈p(ω), g〉 is measurable, for all simple functions gon S. Let h ∈ BL+(S). Then there are simple functions hn such that 0 ≤ hn ↑ h.By the Lebesgue Monotone Convergence Theorem, 〈p(ω), hn〉 → 〈p(ω), h〉 forevery ω ∈ Ω. So ω 7→ 〈p(ω), h〉 is the pointwise limit of measurable functions,hence measurable. For general h ∈ BL(S), we can write h = h+ − h−, and thusω 7→ 〈p(ω), h〉 is the difference of two measurable functions, hence measurable.So p is weakly measurable. Since SBL is separable, p is strongly measurable byPettis’ Theorem.
If p : Ω → S+BL is Bochner integrable with respect to µ ∈ M+(Ω), then ν :=∫
Ωp(ω)dµ(ω) defines an element in SBL. Since S+
BL =M+(S) is a closed convexcone in SBL, ν is in S+
BL.
Proposition 2.6. Let p : Ω → S+BL be Bochner integrable with respect to µ in
M+(Ω), and define ν :=∫
Ωp(ω)dµ(ω). Then∫
S
f dν =∫
Ω
〈p(ω), f〉dµ(ω), (1)
for any bounded measurable f : S → R.
Proof. Step 1. (1) holds for all f ∈ BL(S).We can view f as element of S∗BL. Since p is Bochner integrable with respectto µ, the map Ω→ R : ω 7→ 〈p(ω), f〉 is in L1(µ). So we get by [8, Theorem 6]that ∫
S
f dν = 〈ν, f〉 =⟨∫
Ω
p(ω)dµ(ω), f⟩
=∫
Ω
〈p(ω), f〉dµ(ω).
Step 2. (1) holds for all f = 11C , C ⊂ S closed.Let fn ∈ BL(S) be defined as fn(x) := max(1−nd(x,C), 0). Then fn is boundedby 11, and fn(x)→ 11C(x) for all x ∈ S, so by Lebesgue Dominated ConvergenceTheorem we have that for all ω ∈ Ω, 〈p(ω), fn〉 → 〈p(ω), 11C〉 = [p(ω)](C). Sincefn ∈ S∗BL, ω 7→ 〈p(ω), fn〉 is in L1(µ). Also
‖〈p(ω), fn〉‖ ≤ ‖p(ω)‖TV = ‖p(ω)‖∗BL,
for all ω ∈ Ω and n ∈ N, and by assumption ω 7→ ‖p(ω)‖∗BL is in L1(µ). Henceby the Lebesgue Dominated Convergence Theorem∫
Ω
〈p(ω), fn〉dµ(ω)→∫
Ω
[p(ω)](C)dµ(ω).
By Step 1 ∫Ω
〈p(ω), fn〉dµ(ω) =∫S
fndν,
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for all n ∈ N. And again by the Lebesgue Dominated Convergence Theoremwe can conclude that
∫Sfndν → ν(C). So
∫Ω
[p(ω)](C)dµ(ω) = ν(C) for all Cclosed.Step 3. (1) holds for all bounded measurable f : S → R.Now we want to apply Theorem 2.4. Let H be the vector space of boundedmeasurable functions f : S → R, such that
∫Ω〈p(ω), f〉dµ(ω) =
∫Sfdν. Note
that these expressions are well defined: f is bounded and measurable, so itfollows from Proposition 2.5 that Ω→ R : ω 7→ 〈p(ω), f〉 is in L1(µ).
By Step 2 11C ∈ H for all C ⊂ S closed. Now let fn ∈ H with 0 ≤ fn ↑f ≤ M < ∞, for some function f and some M > 0. Then by the LebesgueMonotone Convergence Theorem, 〈p(ω), fn〉 → 〈p(ω), f〉 for all ω ∈ Ω, and∫Sfndν →
∫Sfdν. Since 〈p(ω), fn〉 is bounded from above by a constant not
depending on n and ω, we can apply the Lebesgue Dominated ConvergenceTheorem to get that∫
Ω
〈p(ω), fn〉dµ(ω)→∫
Ω
〈p(ω), f〉dµ(ω).
Since fn ∈ H we can conclude that∫
Ω〈p(ω), f〉dµ(ω) =
∫Sfdν, hence f ∈ H.
By Theorem 2.4 we obtain that H contains every bounded real-valued Borelmeasurable function.
Corollary 2.7. Let p : Ω → S+BL be Bochner integrable with respect to µ in
M+(Ω). Then [∫Ω
p(ω)dµ(ω)]
(E) =∫
Ω
p(ω)(E)dµ(ω)
for any Borel measurable E ⊂ S.
3 Markov semigroups
We start by introducing the concept of Markov operators.
Definition 3.1. A Markov operator is a map P :M+(S)→M+(S), such that
(MO1) P is additive and R+-homogeneous,
(MO2) ‖Pµ‖TV = ‖µ‖TV for all µ ∈M+(S).
Since M(S)TV is a Banach lattice, condition (MO1) ensures that a Markovoperator P extends to a positive bounded linear operator on M(S)TV given byPµ := P (µ+)− P (µ−). The operator norm of this extension is
‖P‖ = sup‖Pµ‖TV : µ ∈M+(S), ‖µ‖TV ≤ 1 = 1
according to (MO2). Since Id :M(S)TV →M(S)BL is continuous with operatornorm equal to 1, (MO2) implies that P :M(S)TV →M(S)BL is nonexpansiveand an isometry on the positive cone.
A Markov semigroup is a semigroup (P (t))t≥0 of Markov operators.
While strong continuity of (P (t))t≥0 with respect to ‖ · ‖TV is rare, we will seethat strong continuity with respect to ‖ · ‖∗BL is not.
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Thus, we call the Markov semigroup (P (t))t≥0 strongly continuous, when t 7→P (t)µ : R+ → M(S)BL is continuous for each µ ∈ M+(S). Then by linearity,this continuity holds for all µ ∈M(S).
Lemma 3.2. Let (P (t))t≥0 be a Markov semigroup. Then the following areequivalent:
(i) (P (t))t≥0 is strongly continuous,
(ii) 〈P (t)µ, f〉 → 〈µ, f〉 as t ↓ 0 for all µ ∈M+(S), f ∈ Cb(S).
(iii) 〈P (t)µ, f〉 → 〈µ, f〉 as t ↓ 0 for all µ ∈M(S), f ∈ Cb(S).
Proof. (i)⇒ (ii): Follows from Lemma 2.3, since µ, P (t)µ ∈M+(S).(ii)⇒ (iii): Follows from the decomposition µ = µ+ − µ−.(iii)⇒ (i): This is a direct consequence of [9, Theorem 6].
We will show that certain actions of R+ on S provide us with an important classof examples of Markov semigroups.
A semigroup of continuous maps on S is a family of maps (Φt)t≥0, such thatΦt : S → S is continuous, Φt Φs = Φt+s and Φ0 = IdS for all s, t ∈ R+.(Φt)t≥0 is strongly continuous if the map R+ → S : t 7→ Φt(x) is continuous forall x ∈ S.
Proposition 3.3. Let (Φt)t≥0 be a semigroup of continuous maps on S. ThenP (t)µ := µ Φ−1
t defines a Markov semigroup (P (t))t≥0, such that P (t) :M+(S)BL → M+(S)BL is continuous for all t ∈ R+. This Markov semigroupis strongly continuous if and only if (Φt)t≥0 is strongly continuous.
Proof. Let µ ∈M+(S). It is easily verified that P (t)µ ∈M+(S), P (t)P (s)µ =P (t + s)µ for all s, t ∈ R+ and P (0) = Id, and that (P (t))t≥0 satisfies (MO1)-(MO2).
Fix t ∈ R+. Let µn, µ ∈ M+(S) such that ‖µn − µ‖∗BL → 0 as n → ∞. Thenfor all f ∈ Cb(S),
∫Sf dµn →
∫Sf dµ by Lemma 2.3. Hence
〈P (t)µn, f〉 = 〈µn, f Φt〉 → 〈µ, f Φt〉 = 〈P (t)µ, f〉.
Therefore ‖P (t)µn − P (t)µ‖∗BL → 0, hence P (t) : M+(S)BL → M+(S)BL iscontinuous.
Suppose (Φt)t≥0 is strongly continuous and let f ∈ Cb(S). Then f Φt(x) →f(x) as t ↓ 0. Also, |f Φt(x)| ≤ ‖f‖∞11 ∈ L1(µ), hence by the LebesgueDominated Convergence Theorem
〈P (t)µ, f〉 = 〈µ, f Φt〉 → 〈µ, f〉,
so (P (t))t≥0 is strongly continuous by Lemma 3.2.
Now suppose (P (t))t≥0 is strongly continuous and let x ∈ S. Then P (t)δx =δΦt(x) → δx as t ↓ 0. It follows from [17, Lemma 3.5] that for all y, z ∈ S,d(y, z) = 2‖δy−δz‖∗BL
2−‖δy−δz‖∗BL, so
d(Φt(x), x) =2‖P (t)δx − δx‖∗BL
2− ‖P (t)δx − δx‖∗BL
→ 0,
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as t ↓ 0. Hence (Φt)t≥0 is strongly continuous.
In [17, Section 5] it is shown that if, in addition to the conditions above, themaps Φt : S → S are Lipschitz, then the Markov semigroup (P (t))t≥0 canbe extended to a semigroup of bounded linear operators (P (t))t≥0 on SBL.Moreover, (P (t))t≥0 is strongly continuous if (Φt)t≥0 is strongly continuous andlim supt↓0 |Φt|Lip <∞.
4 Restriction to invariant L1-spaces
Let µ ∈ M+(S). For f ∈ L1(µ) we define jµ(f) = f dµ. Then jµ is a linearmap from L1(µ) into M(S).
Lemma 4.1. The following properties hold:
(i) jµ is an isometric embedding of L1(µ) into M(S)TV, i.e. ‖jµ(f)‖TV =‖f‖1 for all f ∈ L1(µ).
(ii) jµ is a continuous embedding of L1(µ) into SBL, with ‖jµ(f)‖∗BL = ‖f‖1for all f ∈ L1
+(µ) and ‖jµ(f)‖∗BL ≤ ‖f‖1 for all f ∈ L1(µ).
The proof is straightforward.
Let P : M+(S) → M+(S) be a Markov operator. Suppose that P leavesjµ(L1
+(µ)) invariant. Then P induces an additive and positively homogeneousmap T : L1
+(µ)→ L1+(µ):
Tf := j−1µ P jµ(f).
Because L1(µ) is a Banach lattice, T extends to a positive bounded linear op-erator on L1(µ), which we will also denote by T , and
‖T‖ = sup‖Tf‖ | f ∈ L1+(µ), ‖f‖ ≤ 1 = 1 (2)
by Lemma 4.1 and (MO2). T will be called the operator (in L1(µ)) induced byP .
At this point we would like to note, that
Lemma 4.2. If µ, ν ∈M+(S) satisfy µ ν, then Pµ Pν.
Proof. There exists f ∈ L1+(ν), such that jν(f) = µ. There are fn ∈ L∞+ (µ)
Furthermore, 0 ≤ jν(fn) ≤ ‖fn‖∞ν. Hence by positivity of P , 0 ≤ Pjν(fn) ≤‖fn‖∞Pν. Therefore Pjν(fn) Pν, hence Pjν(fn) ∈ L1
+(Pν) for all n ∈ N.Because L1
+(Pν) is closed in M+(S)TV, Pjν(f) ∈ L1(Pν) as well, thus Pµ Pν.
Corollary 4.3. P leaves jµ(L1+(µ)) invariant if and only if Pµ µ.
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Proof. Clearly, if P leaves jµ(L1+(µ)) invariant, then in particular Pµ µ.
The proof in the opposite direction follows from Lemma 4.2: if f ∈ L1+(µ), then
0 ≤ jµ(f) µ, hence Pjµ(f) Pµ µ.
Crucial in our approach is the following general topological closed graph theorem(cf. [22], (14.1.2), p.313):
Proposition 4.4. Let f map the topological space S into the topological spaceT . If f is closed and T is compact, then f is continuous.
Moreover, we use the following characterisation of relatively weakly compactsubsets of L1 (e.g. [1], Theorem 5.2.9, p. 109):
Theorem 4.5 (Dunford-Pettis). Let (Ω,Σ, µ) be a σ-finite measure space. Inaddition let F be a bounded set in L1(µ). Then the following conditions on Fare equivalent:
(i) F is relatively weakly compact.
(ii) For every sequence An of disjoint measurable sets
limn→∞
supf∈F
∫An
|f | dµ = 0.
The fundamental result of this section is:
Theorem 4.6. Let (P (t))t≥0 be a strongly continuous Markov semigroup. Letµ ∈ M+(S) be such that jµ(L1
+(µ)) is (P (t))t≥0-invariant. Then the followingstatements are equivalent:
(i) The semigroup (T (t))t≥0 on L1(µ) induced by (P (t))t≥0 is strongly con-tinuous and positive, and consists of isometries on L1
+(µ).
(ii) The map t 7→ P (t)µ is continuous from R+ to M(S)TV.
(iii) There exists τ > 0 such that for any sequence An of disjoint Borel mea-surable subsets of S,
limn→∞
sup0≤t≤τ
P (t)µ(An) = 0. (3)
Proof. (i) ⇒(ii). From Lemma 4.1 it follows that for s, t ∈ R+
‖P (t)µ− P (s)µ‖TV = ‖P (t)jµ(11)− P (s)jµ(11)‖TV = ‖T (t)11− T (s)11‖1.
By assumption t 7→ T (t)11 is continuous from R+ to L1(µ), hence t 7→ P (t)µ iscontinuous from R+ to M(S)TV.
(ii)⇒(iii). For all s, t ∈ R+ we know by Lemma 4.1 that
‖T (t)11− T (s)11‖1 = ‖P (t)µ− P (s)µ‖TV.
By assumption t 7→ P (t)µ is continuous from R+ to M(S)TV, so t 7→ T (t)11 iscontinuous from R+ to L1(µ).
10
Let τ > 0. By continuity the partial orbit T (t)11|0 ≤ t ≤ τ is norm compact,hence weakly compact. According to Theorem 4.5, for any sequence of disjointmeasurable sets An,
0 = limn→∞
sup0≤t≤τ
∫An
|T (t)11|dµ = limn→∞
sup0≤t≤τ
P (t)µ(An).
(iii)⇒(i). Lemma 4.1, Markov operator property (MO2) and (2) yield that eachT (t) is an isometry on L1
+(µ) and ‖T (t)‖ = 1 for all t ≥ 0. We write L1w to
denote the space L1(µ) with the weak topology and (SBL)w to denote SBL withthe weak topology.Step 1. For any step function f , t 7→ T (t)f : [0, τ ] → L1(µ) has a closed graphin [0, τ ]× L1
w.It suffices to prove the statement for f = 11E , the indicator function of a mea-surable E ⊂ S. To that end, observe that the map jµ : L1(µ) → SBL iscontinuous, hence continuous for the weak topologies in L1(µ) and SBL ([4,Theorem VI.1.1]). The map
ψE : [0, τ ]→ SBL : t 7→ P (t)jµ(11E)
is norm continuous, hence continuous for the weak topology in SBL. Thus itsgraph is closed in [0, τ ] × (SBL)w. We conclude that t 7→ T (t)11E must have aclosed graph in [0, τ ]× L1
w.Step 2. j−1
µ (ψE([0, t])) is compact in L1w.
First of all, it is weakly closed by continuity of jµ : L1w → (SBL)w and compact-
ness of ψE([0, t]) in (SBL)w. According to Theorem 4.5 it suffices to show thatfor any sequence of disjoint measurable subsets An of S,
limn→∞
sup0≤t≤τ
∫An
T (t)11Edµ = 0.
We have 11E ≤ 11, thus 0 ≤ T (t)11E ≤ T (t)11 by positivity of T (t). Therefore,∫An
T (t)11Edµ ≤∫An
T (t)11dµ = P (t)µ(An).
Condition (3) now completes Step 2.Step 3. There is a norm-dense subspace D of L1(µ), such that t 7→ T (t)f isnorm-continuous for f ∈ D.Then (T (t))t≥0 is strongly continuous on L1(µ), because ‖T (t)‖ = 1 for all t (e.g.[11, Proposition I.5.3]). The proof of this step mimicks that of [11, TheoremI.5.8] (‘a weakly continuous semigroup in a Banach space is strongly continu-ous’). According to Step 1, Step 2 and Proposition 4.4, t 7→ T (t)f : R+ → L1(µ)is weakly continuous for any step function f ≥ 0. By separability of S and Pet-tis’ Theorem we conclude that for any step function f , t 7→ T (t)f is measurablein the sense of Bochner. It is integrable over [0, τ ], because ‖T (t)f‖1 ≤ ‖f‖1.Thus we can define as Bochner integral in L1(µ):
fr :=1r
∫ r
0
T (t)fdt, 0 < r ≤ τ.
11
Fix r > 0 and let 0 ≤ s ≤ r. Then for any step function f ,
‖T (s)fr − fr‖1 =1r
∥∥∥∥∫ s+r
s
T (t)fdt−∫ r
0
T (t)fdt∥∥∥∥
1
=1r
∥∥∥∥∫ s+r
r
T (t)fdt−∫ s
0
T (t)fdt∥∥∥∥
1
≤ ‖f‖12sr.
Thus ‖T (t)fr − fr‖1 → 0 as s ↓ 0. Because t 7→ T (t)f is weakly continuous,frn→ f weakly as rn ↓ 0. The step functions are norm dense, hence weakly
dense in L1(µ). Norm closure and weak closure agree on convex sets. Therefore
D := spanfr : f step function, 0 < r ≤ τ
is a norm dense subspace of L1(µ).
Not every strongly continuous Markov semigroup which leaves jµ(L1+(µ)) invari-
ant for some µ ∈ M+(S) satisfies one of the equivalent conditions of Theorem4.6, as the following example will show. Let m denote the Lebesgue measure onRn. The diffusion semigroup (Td(t))t≥0 on L1(m) = L1(Rn,m) is given by
Td(t)f(x) :=∫
Rhd(x− y, t)f(y)dm(y), for t > 0,
where the diffusion kernel hd is given by
hd(x, t) = (4πdt)−n/2e−|x|2/4dt.
Let µ ∈M(Rn), then one can show that x 7→ gµ(x) =∫
R hd(x−y, t)f(y)dµ(y) isin L1(m), and hence defines a measure gµdm. We can extend Td(t) to M(Rn),by defining Td(t)µ to be gµdm. Then Td(t) is linear, leaves M+(Rn) invariant,and ‖Td(t)µ‖∗BL ≤ ‖µ‖∗BL for all µ ∈ M(Rn), so Td(t) can be extended to abounded linear operator on RnBL. Moreover, (Td(t))t≥0 is strongly continuousthere. Hence the restriction of (Td(t))t≥0 to M+(Rn) is a strongly continuousMarkov semigroup. Note that Td(t)µ m for every µ ∈M(Rn) and t > 0. Nowlet f ∈ L1(Rn) such that f > 0 almost everywhere, and set µ = f dm+δ0. ThenTd(t)(µ) µ for all t ≥ 0, so (Td(t))t≥0 leaves L1
+(µ) invariant by Corollary4.3. But Td(t)µ ∈ L1(m) for all t > 0, hence t 7→ Td(t)µ cannot be continuousfrom R+ to M(Rn)TV, since L1(m) is closed in M(Rn)TV and µ 6∈ L1(m), socondition (ii) (hence (i) and (iii)) of Theorem 4.6 is not satisfied.
5 Strong continuity for total variation norm
Let (P (t))t≥0 be a strongly continuous Markov semigroup on S. It extendsto a positive semigroup of bounded linear operators on M(S)TV as we haveseen. Typically the latter is not strongly continuous. In this section we will giveseveral characterisations of the closed invariant subspace of M(S)TV on which(P (t))t≥0 is strongly continuous with respect to ‖ · ‖TV, i.e. the space
M(S)0TV := µ ∈M(S) : t 7→ P (t)µ is continuous from R+ to M(S)TV.
12
Our approach is based on that of Gulick et al. [14]. There the following sit-uation is considered: A locally compact group G acts as a group of homeo-morphisms (Φg)g∈G on a locally compact Hausdorff space X, sending x ∈ Xto Φg(x). This induces an action (P (g))g∈G on the Banach space of boundedRadon measures on X, M(X), endowed with total variation norm, given byP (g)µ(E) := µ(Φg−1E). The subspace ofM(X), consisting of measures µ suchthat g 7→ P (g)µ is continuous from G toM(X) is then identified using convolu-tion of certain functions on G with Radon measures on X, and this identificationis used to provide several characterisations of this subspace (see also [20]).
Adopting this approach to our setting is not straightforward: Instead of a groupG as in [14], we consider a semigroup R+, which implies that actions need notbe invertible. Also, in [14] an action of the group on the underlying spaceX is considered, which induces an action on M(X). While we look, moregenerally, at actions of R+ on M(S) directly, that contain those coming froman underlying action on S by Proposition 3.3. Furthermore, in [14] X mustbe locally compact, since measures on X are defined there by constructingcertain functionals on C0(X); in our setting S needs to be a separable completemetric space, but not necessarily locally compact. We can however overcomethese difficulties by using the Banach space SBL and the theory of integratingfunctions with values in S+
BL = M+(S) as developed in Section 2.3 and proveanalogous characterisations of M(S)0
TV as those in [14] and [20].
These characterisations will help in identifying when the restriction of (P (t))t≥0
to invariant L1-spaces is strongly continuous.
Let A be a Banach algebra with multiplication ∗. A net (eα) in A is an approx-imate identity of A, if limα eα ∗ f = f and limα f ∗ eα = f for all f ∈ A. It isa bounded approximate identity if the net is bounded. A Banach space M is aBanach module over A if there exists a bilinear map ∗ : A ×M → M havingthe following properties:
(BM1) (f ∗ g) ∗m = f ∗ (g ∗m) for all f, g ∈ A,m ∈M.
(BM2) ‖f ∗m‖M ≤ ‖f‖A‖m‖M for all f ∈ A,m ∈M.
Proposition 5.1. ([14, Corollary 2.3]) Let A be a Banach algebra with boundedapproximate identity (eα). If M is a Banach module over A, then A ∗M :=a ∗m : a ∈ A,m ∈ M is a closed subspace of K. In particular, for m ∈ M ,m ∈ A ∗M if and only if limα eα ∗m = m.
The latter characterisation of elements in A∗M makes clear that A∗M is indeeda vector subspace of M .
Proposition 5.2. The Banach space L1(R+) is a commutative Banach algebrawith multiplication defined by convolution:
f ∗ g(t) :=∫ t
0
f(t− s)g(s) ds,
with bounded approximate identity (en) given by en = n11[0, 1n ].
The proof is straightforward, observing that L1(R+) is canonically contained asclosed subspace in the commutative Banach algebra L1(R) with convolution.
13
For a strongly continuous Markov semigroup (P (t))t≥0, t 7→ P (t)µ,R+ → SBL
is continuous for each µ ∈ M(S) (though P (t) :M(S)BL →M(S)BL need notbe continuous) and
‖P (t)µ‖∗BL ≤ ‖P (t)µ‖TV ≤ ‖µ‖TV,
Thus P (·)µ ∈ Cb(R+,SBL) and we can define for f ∈ L1(R+) and µ ∈M(S)
f ∗P µ :=∫
R+
f(s)P (s)µds
as Bochner integral in SBL. Clearly (f, µ) 7→ f ∗P µ is a bilinear map fromL1(R+) ×M(S) to SBL. Because S+
BL is closed and convex in SBL, f ∗P µ ∈S+
BL =M+(S), when f ∈ L1+(R+) and µ ∈M+(S). By writing f ∈ L1(R+) and
µ ∈ M(S) as difference of positive and negative parts f± and µ± respectively,it follows that
by using (4) and the fact that M(S) and L1(R+) are L-spaces.
Put L1(R+) ∗P M(S) := f ∗ µ : f ∈ L1(R+), µ ∈M(S).
Then we have, by Proposition 5.2, Proposition 5.4 and Proposition 5.1, thefollowing result:
Corollary 5.5. M(S)TV is a Banach module over L1(R+). In particular,L1(R+) ∗P M(S) is a non-trivial closed subspace of M(S)TV.
This closed subspace equals the subspace of strong continuity of P (t) with re-spect to ‖ · ‖TV:
Theorem 5.6. For µ ∈M(S) the following are equivalent:
(i) µ ∈M(S)0TV, i.e. t 7→ P (t)µ : R+ →M(S)TV is continuous.
(ii) µ ∈ L1(R+) ∗P M(S).
(iii) If E is a Borel set in S such that P (t)µ(E) = 0 for almost every t ∈ [0,∞),then µ(E) = 0.
15
(iv) There exists ν ∈M+(S)0TV such that jν(L1(ν)) is
(P (t))t≥0-invariant and µ ∈ jν(L1(ν)).
Proof. (i)⇒ (ii): Let µ ∈ M(S)0TV. By Proposition 5.1 it is sufficient to show
that en ∗ µ → µ. Let ε > 0. Since t 7→ P (t)µ : R+ → M(S)TV is continuous,there exists an N ∈ N, such that ‖P (t)µ−µ‖TV ≤ ε for all t ∈ [0, 1
N ]. For n ∈ N
en ∗ µ− µ = n
∫ 1n
0
P (t)µ− µdt
is defined as Bochner integral in SBL.
By continuity, t 7→ P (t)µ − µ : [0, 1n ] → M(S)TV is strongly measurable and
bounded, hence Bochner integrable, so we can also view the integral n∫ 1
n
0P (t)µ−
µds as a Bochner integral in M(S)TV. Since M(S)TV embeds continuously inSBL, the two integrals are the same.
Moreover,
‖en ∗P µ− µ‖TV ≤ n∫ 1
n
0
‖P (t)µ− µ‖TV dt ≤ ε,
for all n ≥ N .
(ii)⇒ (i): Let µ = f ∗P ν ∈ L1(R+) ∗P M(S). Let t, s ≥ 0. According toProposition 5.3 and Proposition 5.4,
‖P (t)µ− µ‖TV = ‖(R+(t)f) ∗P ν − f ∗P ν‖TV
≤ ‖R+(t)f − f‖1‖ν‖TV.
Since (R+(t))t≥0 is strongly continuous on L1(R+), ‖R+(t)f − f‖1 → 0 as t ↓ 0and thus ‖P (t)µ− µ‖TV → 0. So µ ∈M(S)0
TV.
Thus from now on we can identify M(S)0TV with L1(R+) ∗P M(S).
(i)⇒ (iii): Let µ ∈M(S)0TV and let E be a Borel set in S. Then t 7→ P (t)µ(E)
is continuous, hence if P (t)µ(E) = 0 for almost every t ∈ [0,∞), then µ(E) = 0.
(iii)⇒ (iv) Let f ∈ L1(R+), such that f(t) > 0 for almost every t ∈ [0,∞).Define ν = f ∗P |µ|. Suppose ν(E) = 0 for a Borel set E in S, then P (t)|µ|(E) =0 for almost every t ∈ [0,∞). By positivity of P (t), |P (t)µ|(E) ≤ P (t)|µ|(E) = 0for almost every t ∈ [0,∞), hence µ(E) = 0 and µ ν. Furthermore, byProposition 5.3,
P (t)ν(E) =∫
R+
f(s)P (t+ s)|µ|(E) ds = 0,
hence P (t)ν ν. According to Corollary 4.3, (P (t))t≥0 leaves jν(L1(ν)) invari-ant, and µ ∈ jν(L1(ν)).
(iv)⇒(i): Since ν ∈ L1(R+) ∗P M(S) =M(S)0TV, t 7→ P (t)ν : R+ →M(S)TV
is continuous. Then Theorem 4.6 implies that the semigroup (T (t))t≥0 in L1(ν)induced by (P (t))t≥0 is strongly continuous. By assumption there is an f ∈L1(ν) such that jν(f) = µ. Then
‖P (t)µ− µ‖TV = ‖T (t)f − f‖1 → 0,
as t ↓ 0.
16
The following theorem gives some useful conditions for showing that µ ∈M(S)0TV.
Theorem 5.7. Let µ ∈M(S). Then the following are equivalent:
(i) µ ∈M(S)0TV.
(ii) For all compact K in S, t 7→ P (t)µ(K) is continuous.
(iii) If K in S compact and P (t)µ(K) = 0 for almost every t ∈ [0,∞), thenµ(K) = 0.
(iv) There is a ν ∈M(S)0TV such that µ ν.
Proof. (i)⇒ (ii): Since µ ∈ M(S)0TV, t 7→ P (t)µ(E) is continuous for all Borel
sets E in S.
(ii)⇒ (iii): Let K in S be compact, such that P (t)µ(K) = 0 for almost everyt ∈ [0,∞). Then, by continuity of t 7→ P (t)µ(K), µ(K) = 0.
(iii)⇒ (iv): Let f ∈ L1(R+), such that f(t) > 0 for almost every t ∈ [0,∞).Define ν := f ∗P |µ|. Let K in S be compact, such that ν(K) = 0, thenP (t)|µ|(K) = 0 for almost every t ∈ [0,∞). By positivity of P (t), |P (t)µ|(K) ≤P (t)|µ|(K) = 0 for almost every t ∈ [0,∞), hence µ(K) = 0. Thus µ ν byLemma 2.1.
(iv)⇒ (i): Let f ∈ L1(R+), such that f(t) > 0 for almost every t ∈ [0,∞).Define ρ := f ∗P |ν| ∈ M+(S)∩L1(R+) ∗PM(S). Now, let E be a Borel set inS such that ρ(E) = 0. Then P (t)|ν|(E) = 0 for almost every t ∈ [0,∞). Notethat |ν|(E) = 0 if and only if ν(F ) = 0 for all Borel sets F ⊂ E. Let F ⊂ E beBorel, then by positivity of P (t),
|P (t)ν(F )| ≤ P (t)|ν|(F ) = 0,
for almost every t ∈ [0,∞). Since ν ∈ L1(R+) ∗P M(S), t 7→ P (t)ν(F ) iscontinuous, so ν(F ) = 0. So µ ν ρ.
Also, by Proposition 5.3, we have for every s ≥ 0
P (s)ρ(E) =∫
R+
f(t)P (t+ s)|ν|(E) dt = 0,
since P (t)|ν|(E) = 0 for every t ≥ 0. So P (t)ρ ρ for all t ≥ 0, and µ ρ.By Corollary 4.3 (P (t)0t≥0 leaves jρ(L1(ρ)) invariant, and µ ∈ jρ(L1(ρ)). Nowwe can apply Theorem 5.6.
Corollary 5.8. Let µ ∈ M+(S). If there is a τ > 0 such that µ P (t)µ forall t ∈ [0, τ ], then µ ∈M(S)0
TV.
Proof. Let E ⊂ S be measurable such that P (t)µ(E) = 0 for almost everyt ∈ [0,∞). Then there is a t ∈ [0, τ ] such that P (t)µ(E) = 0, and then µ(E) = 0,since µ P (t)µ. Hence µ ∈M(S)0
TV by Theorem 5.6.
An important consequence of the characterisations in Theorem 5.6 is:
Proposition 5.9. M(S)0TV is dense in M(S)BL, hence in SBL.
17
Proof. Let µ ∈ M(S) and ε > 0. Then there is a τ > 0 such that ‖P (t)µ −µ‖∗BL < ε for all t ∈ [0, τ ]. By Theorem 5.6 en ∗P µ ∈M(S)0
TV.
‖en ∗P µ− µ‖∗BL = n‖∫ 1
n
0
P (t)µ− µdm‖∗BL
≤ n
∫ 1n
0
‖P (t)µ− µ‖∗BL dm < ε,
for all t ∈ [0, τ ]. So ‖en ∗P µ−µ‖∗BL → 0, andM(S)0TV is dense inM(S)BL.
However, whenever the Markov semigroup arises from an underlying semigroupof continuous maps on S, M(S)0
TV cannot be too large:
Proposition 5.10. Let (Φt)t≥0 be a strongly continuous semigroup of continu-ous maps on S, and let (P (t))t≥0 be the associated strongly continuous Markovsemigroup. Then M(S)0
TV =M(S) if and only if Φt = Id for every t ∈ R+.
Proof. Suppose Φt = Id for every t ∈ R+. Then P (t)µ = µ for every t ∈ R+
as t ↓ 0. Hence there is a τ > 0 such that δΦt(x) = δx for all t ∈ [0, τ), and thenby continuation δΦt(x) = δx for all t ∈ R+, so Φt(x) = x for all t ∈ R+.
However, there do exist non-trivial strongly continuous Markov semigroups(P (t))t≥0 such that M(S)0
TV = M(S); in [29, Section 5] a C0-semigroup onM(Ω)TV, with (Ω,Σ) a general measurable space, is constructed, which undercertain conditions is a Markov semigroup.
6 Decomposition of the space of measures
6.1 Absolute continuous and singular measures
For µ ∈ M(R), define µt(E) := µ(E − t), t ∈ R. It is a classical result byPlessner [26] that ‖µt − µ‖TV → 0 as t → 0 if and only if µ is absolutelycontinuous with respect to the Lebesgue measure m. Then the Lebesgue-Radon-Nikodym Decomposition Theorem implies that every µ inM(R) can be uniquelydecomposed into µa +µs, where µa ∈ L1(R,m), and µs is singular with respectto m.
We can translate this to our setting: let Φt(x) = x + t, then (Φt)t∈R defines astrongly continuous group of continuous mappings Φt : R → R. This defines astrongly continuous Markov group (PΦ(t))t∈R, by PΦ(t)µ = µ Φ−1
t , by Propo-sition 3.3. Note that we only formulated Proposition 3.3 for semigroups, butit can easily adapted for groups. Plessner’s result implies that the subspace ofstrong continuityM(R)0
TV equals L1(R), and every µ ∈M(R) can be uniquelydecomposed into µa + µs, where µa ∈M(R)0
TV and µs is singular with respectto every ν ∈M(R)0
TV. We will generalise this decomposition in our setting.
18
As in the previous section we assume (P (t))t≥0 is a strongly continuous Markovsemigroup on S, such that P (t) : M+(S)BL → M+(S)BL is continuous for allt > 0.
Proposition 6.1. M(S)0TV is a projection band in M(S)TV.
Proof. We first show that M(S)0TV is an ideal. Let µ, ν ∈ M(S) such that
0 ≤ |µ| ≤ |ν| and ν ∈ M(S)0TV. Then |ν| ∈ M(S)0
TV by Theorem 5.7. Sinceµ |ν|, µ ∈M(S)0
TV, again by Theorem 5.7. Hence M(S)0TV is a closed ideal
in M(S)TV, hence a projection band by Theorem 2.2.
So we can writeM(S) =M(S)0
TV ⊕ (M(S)0TV)⊥, (5)
by Theorem [21, Theorem 1.2.9].
We will show that (M(S)0TV )⊥ =M(S)sTV, where
M(S)sTV := µ ∈M(S) : µ+ ⊥ P (t)µ+, µ− ⊥ P (t)µ− for almost every t ≥ 0.
Our approach is based on that by Liu and Van Rooij [20].
Proposition 6.2. Let µ ∈M(S). Then the following are equivalent:
(i) µ ∈M(S)sTV.
(ii) µ ⊥ ν for every ν ∈M(S)0TV
(iii) For all ν ∈M(S), µ ⊥ P (t)ν for almost every t ∈ [0,∞).
Proof. (i)⇒ (ii): Let ν ∈ M(S)0TV, then |ν| ∈ M(S)0
TV by Theorem 5.7. Bythe Lebesgue-Radon-Nikodym Theorem, there are unique µ+
a , µ+s ∈ M(S)+,
such that µ+ = µ+a + µ+
s , µ+a |ν| and µ+
s ⊥ |ν|. Then µ+a ∈ M(S)0
TV
by Theorem 5.7. By assumption, µ+ ⊥ P (t)µ+ for almost every t ∈ [0,∞).Suppose µ+ ⊥ P (t)µ+, then there is a Borel set U , such that µ+(E) = µ+(E∩U)and P (t)µ+(U) = 0 for all Borel sets E. So
0 ≤ µ+a (E\U) ≤ µ+(E\U) = 0,
hence µ+a (E) = µ+
a (E ∩ U) for all Borel sets E, and
0 ≤ P (t)µ+a (U) ≤ P (t)µ+(U) = 0,
so P (t)µ+a ⊥ µ+
a .
Hence µ+a ⊥ P (t)µ+
a for almost every t ∈ [0,∞). µ+a ⊥ is a band in M(S)TV,
hence closed. Since t 7→ P (t)µ+a : R+ → M(S)TV is continuous, µ+
a ∈ µ+a ⊥,
hence µ+a = 0. This implies that µ+ = µ+
s , so µ+ ⊥ |ν|, and therefore µ+ ⊥ ν.
In a similar way we can prove that µ− ⊥ ν, hence µ ⊥ ν.
(ii)⇒ (iii): Let ν ∈ M(S) and define ρ := f ∗P |ν| ∈ L1(R+) ∗P M(S), wheref ∈ L1(R+), such that f(t) > 0 for almost every t ∈ [0,∞). Then ρ ∈M(S)0
TV
by Theorem 5.6. Then µ ⊥ ρ, hence there is a Borel set U ⊂ S, such thatµ(E) = µ(E ∩ U) and ρ(U) = 0 for all Borel sets E in S. Thus P (t)|ν|(U) = 0for almost every t ∈ [0,∞). Then positivity of (P (t))t≥0 implies that for almost
19
every t ∈ [0,∞), |P (t)ν|(U) = 0, hence |P (t)ν| ⊥ µ. So P (t)ν ⊥ µ for almostevery t ∈ [0,∞).
(iii)⇒ (i): By assumption, µ ⊥ P (t)µ+ and µ ⊥ P (t)µ− for almost everyt ∈ [0,∞). Hence |µ| ⊥ P (t)µ+ and |µ| ⊥ P (t)µ−, so µ+ ⊥ P (t)µ+ andµ− ⊥ P (t)µ− for almost every t ∈ [0,∞).
Corollary 6.3. M(S)sTV = (M(S)0TV)⊥.
This implies that M(S)sTV is a projection band by [21, Proposition 1.2.7].
As in [20] we call µ ∈ M(S) absolutely continuous with respect to (P (t))t≥0
if µ ∈ M(S)0TV and singular with respect to (P (t))t≥0 if µ ∈ M(S)sTV. This
terminology is based on the fact that µ ∈ M(S)0TV if and only if there is a
ν ∈ M(S)0TV such that µ |ν| by Theorem 5.7, and µ ∈ M(S)sTV if and only
if µ and ν are singular for every ν ∈M(S)0TV by Theorem 5.6.
An immediate consequence of (5) and Corollary 6.3 is the following:
Proposition 6.4. Every µ ∈ M(S) has a unique decomposition µ = µa + µs,with µa ∈M(S)0
TV, and µs ∈M(S)sTV.
We denote the band projections on M(S)0TV and M(S)sTV by P0 and Ps re-
spectively. Then P0, Ps are positive bounded linear operators onM(S)TV, with‖P0‖ ≤ 1 and ‖Ps‖ ≤ 1, and P0µ = µa, Psµ = µs.
While M(S)0TV is invariant under (P (t))t≥0, M(S)sTV need not be, as the fol-
lowing example shows: Let S = R+ with euclidean metric. Define Φt(x) =max(x − t, 0), for t, x ∈ R+. Then (Φt)t≥0 is a strongly continuous semigroupof continuous maps on S, hence it defines, by Proposition 3.3, a strongly con-tinuous Markov semigroup (P (t))t≥0 given by P (t)µ := µ Φ−1
t . Let x > 0,then clearly δx ⊥ P (t)δx for all t > 0, hence δx ∈M(S)sTV. However, for t ≥ x,P (t)δx = δ0, and δ0 is in M(S)0
TV, and not in M(S)sTV, since P (t)δ0 = δ0 forall t ∈ R+.
For each µ ∈ M(S), we can define d(µ,M(S)0TV) to be the distance of µ
to M(S)0TV with respect to ‖ · ‖TV. Clearly, µ ∈ M(S)0
TV if and only ifd(µ,M(S)0
TV) = 0.
Lemma 6.5. Let µ ∈M(S). Then d(µ,M(S)0TV) = ‖µs‖TV.
Lemma 6.6. Let µ ∈M(S). The function t 7→ ‖PsP (t)µ‖TV is non-increasing.
20
Proof. It suffices to show that ‖PsP (t)µ‖TV ≤ ‖Psµ‖TV for all t ∈ R+.
Let 0 ≤ t. First assume µ ∈ M+(S), then 0 ≤ µa ≤ µ. Since M(S)0TV is
invariant under P (t), P0P (t)µa = P (t)µa, hence
0 ≤ P (t)µa = P0P (t)µa ≤ P0P (t)µ.
Then
0 ≤ PsP (t)µ = P (t)µ− P0P (t)µ ≤ P (t)µ− P (t)µa = P (t)µs,
hence‖PsP (t)µ‖TV ≤ ‖P (t)µs‖TV ≤ ‖µs‖TV. (6)
Now let µ = µ+ − µ− ∈ M(S). Then Psµ+ ⊥ Psµ
−, which implies that‖Psµ‖TV = ‖Psµ+‖TV + ‖Psµ−‖TV. By (6)
‖PsP (t)µ‖TV ≤ ‖PsP (t)µ+‖TV + ‖PsP (t)µ−‖TV
≤ ‖Psµ+‖TV + ‖Psµ−‖TV = ‖Psµ‖TV.
6.2 A Wiener-Young type theorem
Wiener and Young ([30]) extended the result by Plessner by showing that forall µ ∈ M(R), lim supt→0 ‖µt − µ‖TV = 2‖µs‖TV, where µs is the singularcomponent of µ with respect to the Lebesgue measure.
We can generalise this result to the Markov semigroups with conditions as be-fore. It has been generalised in several other directions: see for instance [23, 24]for a generalisation in the setting of dual semigroups of positive C0-semigroupson Banach lattices.
Theorem 6.7. Let µ ∈M(S). Then lim supt↓0 ‖P (t)µ− µ‖TV = 2‖µs‖TV.
Proof. Step 1. lim supt↓0 ‖P (t)µ− µ‖TV = lim supt↓0 ‖P (t)µs‖TV + ‖µs‖TV forall µ ∈M(S).Clearly ‖P (t)µa − µa‖TV → 0. This implies that lim supt↓0 ‖P (t)µ − µ‖TV =lim supt↓0 ‖P (t)µs − µs‖TV. By Proposition 6.2, P (t)µs ⊥ µs for almost everyt ∈ [0,∞). Hence for these t ‖P (t)µs − µs‖TV = ‖P (t)µs‖TV + ‖µs‖TV, hence
lim supt↓0
‖P (t)µs − µs‖TV ≥ lim supt↓0
‖P (t)µ‖TV + ‖µs‖TV.
Furthermore, ‖P (t)µs−µs‖TV ≤ ‖P (t)µs‖TV +‖µs‖TV for all t ∈ [0,∞), hencethe statement holds.Step 2. lim supt↓0 ‖P (t)µ‖TV = ‖µ‖TV for all µ ∈M(S).Let ε > 0. Since
‖µ‖TV = sup|∫S
f dµ| : f ∈ Cb(S), ‖f‖∞ ≤ 1,
21
there is an f ∈ Cb(S) with ‖f‖∞ ≤ 1 and |‖µ‖TV −∫Sf dµ| < ε
2 . By strongcontinuity of (P (t))t≥0 and Lemma 3.2 there exists a τ > 0, such that
|〈P (t)µ, f〉 − 〈µ, f〉| < ε
2for all t ∈ [0, τ).
Thus for t ∈ [0, τ) we can conclude
‖P (t)µ‖TV ≥ |〈P (t)µ, f〉| ≥ ‖µ‖TV − ε,
which implies that lim supt↓0 ‖P (t)µ‖TV ≥ ‖µ‖TV. By (MO2)
lim supt↓0
‖P (t)µ‖TV ≤ ‖µ‖TV,
which completes the proof.
A Auxiliary results from measure theory
Let (Ω,Σ, µ) be a measure space and X a Banach space. Let f be an X-valuedfunction on Ω. It is called a simple function if it is constant on each of afinite number of pairwise disjoint measurable sets En, such that
⋃nEn = Ω
and µ(ω : ‖f(ω)‖ > 0
)< ∞. f is weakly measurable if for each x∗ ∈ X∗,
Ω → R : ω 7→ x∗(f(ω)) is measurable. It is strongly µ-measurable if thereexists a sequence of simple functions that converges pointwise in X, µ-almosteverywhere. f is called essentially separably valued if there exists a µ-null setN , such that f(Ω \N) is separable. Weak and strong measurability are relatedby Pettis’ Theorem (e.g. [8, 15, 25]).
Theorem A.1 (Pettis’ Measurability Theorem). Let (Ω,Σ, µ) be a σ-finitemeasure space, X a Banach space and f an X-valued function on Ω. Thefollowing are equivalent:
(i) f is strongly µ-measurable,
(ii) f is weakly µ-measurable and essentially separably valued,
(iii) There exists µ-null sets N and N ′, N ⊂ N ′, and a sequence of simplefunctions fn : Ω→ X, such that fn(Ω) ⊂ f(Ω\N)∪0 and fn(ω)→ f(ω)as n→∞ for all ω ∈ Ω \N ′. Furthermore, for all ω ∈ Ω \N and n ∈ N,either ‖fn(ω)− f(ω)‖ < 1
n or fn(ω) = 0.
If µ is finite, then the range of each of the simple functions fn may taken to becontained in f(Ω \N).
The usual statement involves only part (i ) and (ii ). When examining the prooffound in [15, Theorem 3.5.3, p. 73], however, one finds that one actually exhibitsa sequence of simple functions as stated in part (iii ), which we will use in theproof of Theorem A.3 below.
Lemma A.2. Let X and Y be normed vector spaces and Γ a convex cone in X.If φ : Γ→ Y is continuous at 0 and positively homogeneous, then φ is bounded,i.e. there exists an C ≥ 0 such that ‖φ(x)‖Y ≤ C‖x‖X for all x ∈ Γ.
22
Proof. It suffices to show that there exists an M ≥ 0 such that ‖φ(x)‖Y ≤ Mfor all x ∈ Γ, ‖x‖X ≤ 1. Suppose that there is no such M . Then there arexn ∈ Γ such that ‖xn‖X ≤ 1 and φ(xn) ≥ 2n. Define zn := xn
2n ∈ Γ. Then‖zn‖X ≤ 1
2n , so zn → 0 in Γ as n→∞ and φ(zn)→ φ(0) = 0 by continuity ofφ. This contradicts φ(zn) ≥ 1.
Theorem A.3. Let (Ω,Σ, µ) be a finite measure space, X a Banach space andf : Ω→ X a µ-Bochner integrable function. Let Γ ⊂ X be a closed convex cone,such that f(ω) ∈ Γ for µ-almost every ω ∈ Ω. Let G : Γ → X be continuous,positively homogeneous and additive. Then
G
∫Ω
f dµ =∫
Ω
G fdµ.
Proof. Let N1 ∈ Σ be a µ-null set, such that f(Ω \ N1) ⊂ Γ. Define f =11Ω\N1 · f , then f equals f µ-almost everywhere, and f(Ω) ⊂ Γ. So without lossof generality we can assume that f(Ω) ⊂ Γ.
f is strongly µ-measurable, so by Theorem A.3 there exist µ-null sets N,N ′
and simple functions fn : Ω→ X such that the conditions in Theorem A.1.(iii)are satisfied. Then for every ω ∈ Ω \ N ′ and every n ∈ N, ‖fn(ω)‖ < 1
n +‖f(ω)‖ ≤ 1 + ‖f(ω)‖. Since f is µ-Bochner integrable and µ is finite, 1 +‖f(·)‖ is in L1(µ). Hence by the vector-valued Dominated Convergence Theorem∫
Ωfn dµ →
∫Ωf dµ. Since fn(Ω) ⊂ Γ for every n ∈ N, f(Ω) ⊂ Γ and Γ is a
closed convex cone, it follows from [8, Corollary II.8] that∫
Ωfndµ,
∫Ωfdµ ∈ Γ.
So G∫
Ωfndµ→ G
∫Ωf dµ.
Let h : Ω → X be a simple function with h(Ω) ⊂ Γ. We can write h =∑mk=1 11Ek
xk, with the Ek ∈ Σ pairwise disjoint, and xk ∈ Γ. Then G h is asimple function. Since G is additive and positively homogeneous, G
∫Ωh dµ =∫
ΩG h dµ. G is also continuous, hence it is bounded by Lemma A.2, so there
exists a C > 0 such that ‖Gx‖ ≤ C‖x‖ for all x ∈ Γ. Thus for every ω ∈ Ω\N ′,‖G fn(ω)‖ ≤ C‖fn(ω) ≤ C(1 + ‖f(ω)‖) and G fn(ω) → G f(ω). By thevector-valued Dominated Convergence Theorem,
∫ΩG fn dµ →
∫ΩG f dµ.
Hence
G
∫Ω
f dµ = limn→∞
G
∫Ω
fn dµ = limn→∞
∫Ω
G fn dµ =∫
Ω
G fdµ.
23
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