VECTOR INTEGRALS AND A POINTWISE MEAN ERGODIC THEOREM FOR TRANSITION FUNCTIONS

(1)

VECTOR INTEGRALS AND A POINTWISE MEAN ERGODIC THEOREM

FOR TRANSITION FUNCTIONS

RADU ZAHAROPOL

Our goal is to discuss the use of a certain vector integral in order to extend the weak* and pointwise mean ergodic theorems for equicontinuous Markov-Feller pairs to equicontinuous Feller transition functions.

AMS 2000 Subject Classification: Primary 28B05; Secondary 26A42, 60J35.

Key words: Bochner integral, Markov operator, mean ergodic theorem, pointwise convergence, transition function, weak* convergence.

PROLOGUE

The main ideas of the paper can be told in the language of fairy tales: We consider Cinderella (in Romanian we call her “Cenu¸s˘areasa”) and a handsome prince (in Romanian, “F˘at-Frumos”). Cinderella (the pointwise integral that will be defined in the paper) is a modest girl without education (knows almost no measure theory, let alone ergodic theory), but she is easygoing and exists under almost any conditions. By contrast, the prince (the Bochner integral) has a very solid measure theoretical foundation and (thanks to Dunford and Schwartz [3]) is familiar with ergodic theory; however, the prince is spoiled, and exists under rather sophisticated conditions. Together, Cinderella and the prince can help us deal with two ergodic theorems. And then, of course, they will live happily ever after (again, in Romanian we say “Am ˆınc˘alecat pe-o ¸sea

¸si v-am spus povestea a¸sa.”).

1. INTRODUCTION

In [6] and [8] we have obtained various results concerning transition probabilities: an extension of an ergodic decomposition that goes back to Kryloff, Bogoliouboff, Beboutoff and Yosida, results on supports of various types of invariant probabilities, a weak* mean ergodic theorem, and a pointwise mean

REV. ROUMAINE MATH. PURES APPL.,53(2008),1, 63–78

(2)

ergodic theorem. In recent years we have extended all the above-mentioned results to transition functions, and currently we are writing the final form of a small monograph on transition functions that will include these extensions. As expected, the proofs of the extensions are more sophisticated than the proofs of the corresponding results for transition probabilities as one encounters various difficulties due to the transition function setting. Our goal in this note is to show how to overcome one such difficulty when extending the pointwise and the weak* mean ergodic theorems. In doing so, we will discuss a vector integral; it seems that the integral has never been mentioned explicitly in the literature, but, in all likelihood, it has been used implicitly earlier.

Let (X, d) be a locally compact separable metric space and let B(X) be the σ-algebra of all Borel subsets ofX.

As usual, we say that a mapP :X×B(X)→Ris atransition probability if the following two conditions are satisfied.

(i) For everyx∈X, the mapµx:B(X)→Rdefined byµx(A) =P(x, A) for every A∈ B(X) is a probability measure.

(ii) For every A ∈ B(X), the function g_A :X → R defined by g_A(x) = P(x, A) for everyx∈X is Borel measurable.

Let B_b(X) be the Banach lattice (under the sup (uniform) norm and the pointwise order) of all real-valued bounded Borel measurable functions defined on X, and let M(X) be the Banach lattice of all real-valued signed Borel measures onX, the norm onM(X) being the total variation norm, and the order onM(X) being the usual one (for the definition and a comprehensive treatment of Banach lattices, see the classical monograph of Schaefer [5]).

Given a transition probabilityP on (X, d), we can define two operators S :B_b(X)→B_b(X) and T :M(X)→ M(X) by

(1.1) Sf(x) =

Z

X

f(y)P(x,dy)

for every f ∈B_b(X) andx ∈X, where P(x,dy) stands for dµ_x(y), µ_x being the probability measure defined by P and x ∈ X that appears at (i) in the definition of a transition probability, and by

(1.2) T µ(A) =

Z

X

P(x, A) dµ(x) for every µ∈ M(X) and A∈ B(X).

It is easy to see that S and T are well-defined in the sense that the functionSf obtained using (1.1) belongs toBb(X) wheneverf ∈Bb(X), and that if T µis given by (1.2), then T µbelongs to M(X) whenever µ∈ M(X).

Also easy to see is the fact that S and T are linear positive contractions

(3)

of Bb(X) and M(X), respectively. We call the pair (S, T) the Markov pair defined by P. For details on Markov pairs, see [8].

LetCb(X) be the Banach sublattice of Bb(X) of all real-valued continuous bounded functions defined on X.

A pair (S, T) is called aMarkov-Feller pair ifSf belongs toC_b(X) wheneverf ∈C_b(X). If (S, T) is a Markov-Feller pair, the corresponding transition probability P is called a Feller transition probability. For details on Markov- Feller pairs, see [6] and [8].

A family (P_t)_t∈[0,+∞)of transition probabilities defined on (X, d) is called a transition function if it has the property that

(1.3) Ps+t(x, A) =

Z

X

Ps(y, A)Pt(x,dy)

for every s∈ [0,+∞), t∈[0,+∞), A∈ B(X) andx ∈X. Note that (1.3) is the familiar Chapman-Kolmogorov equation.

Let (Pt)t∈[0,+∞) be a transition function. For each t ∈ [0,+∞) we can construct the Markov pair (St, Tt) defined by Pt. We call the family ((S_t, T_t))_t∈[0,+∞) constructed in this way the family of Markov pairs defined (or generated) by (P_t)_t∈[0,+∞). It can be shown that the families (S_t)_t∈[0,+∞) and (T_t)_t∈[0,+∞) are semigroups of operators (on B_b(X) and M(X), respectively).

We say that (Pt)t∈[0,+∞) is a Feller transition function if (St, Tt) is a Markov-Feller pair for every t ∈ [0,+∞). In this case, we also say that (S_t)_t∈[0,+∞) is aFeller semigroup of operators.

We say that (P_t)_t∈[0,+∞) satisfies thestandard measurability assumption (s.m.a.) if for everyA∈ B(X), the map (t, s)7→P_t(x, A), (t, x)∈[0,+∞)×X, is jointly measurable with respect to t and x (that is, the map is measurable with respect to the product σ-algebra L([0,+∞))⊗ B(X), where L([0,+∞)) is the σ-algebra of all Lebesgue measurable subsets of [0,+∞)).

LetC₀(X) be the Banach sublattice ofC_b(X) (and, of course, ofB_b(X)) of all real-valued continuous functions on X that vanish at infinity.

It can be shown that a Feller transition function which isC₀(X)-strongly continuous satisfies the s.m.a.

We say that (Pt)_t∈[0,+∞) (or (St)_t∈[0,+∞), or ((St, Tt))_t∈[0,+∞)) isC0(X)- strongly continuous if for every f ∈ C0(X), the map t 7→ Stf, t ∈ [0,+∞), is continuous with respect to the topology of uniform convergence on B_b(X) and the usual standard topology on [0,+∞).

We say that (P_t)_t∈[0,+∞) (or (S_t)_t∈[0,+∞), or ((S_t, T_t))_t∈[0,+∞)) isC₀(X)- equicontinuous if for everyf ∈C₀(X), for every convergent sequence (x_n)n∈N

of elements of X, and for every ε ∈ R, ε > 0, there existsnε ∈N such that

|S_tf(x_n) −S_tf(x)| < ε for every t ∈ [0,∞) and for every n ≥ n_ε, where x= limn→∞xn.

(4)

Now, assume that (Pt)t∈[0,∞) satisfies the s.m.a. It can be shown that for every f ∈Bb(X) and everyx∈X, the map t7→Stf(x),t∈[0,∞) is measurable. Since the map is also bounded, the integral Rs

0 S_tf(x) dt exists and is a real number for every s ∈[0,∞). Hence, it makes sense to try to single out a class of transition functions (P_t)_t∈[0,∞) that satisfy the s.m.a. and have the property that lim

s→∞

1 s

Rs

0 S_tf(x) dt exists for every f ∈ C₀(X) and x∈ X, thus extending the pointwise ergodic theorem for Feller transition probabilities that we proved in [6], Section 4.3, Corollary 4.3.2. However, at least for the time being, the only manner in which we can obtain such an extension is to obtain first an extension of the weak* mean ergodic theorem for transition probabilities discussed in Theorem 4.3.1 of [6] to transition functions. In order to obtain such an extension, we have to assign a meaning toRs

0 Ttµdtfor every µ ∈ M(X) and s∈[0,+∞) in order to be able to obtain conditions that in- sure the existence of w*- lim

s→+∞

1 s

Rs

0 Ttµdtfor everyµ∈ M(X), where w*-lim stands for the limit in the weak* topology ofM(X). Ideally, we would like the integrals Rs

0 Ttµdt,s≥0, to be Bochner integrals (for details on the Bochner integral, see Dinculeanu’s comprehensive monograph [2] on vector and stochastic integrals and Appendix E of the excellent textbook in measure theory of Cohn [1]) because in such a case we could try to use results from the first volume of Dunford and Schwartz’s monograph [3] to study conditions for the existence of w*- lim

s→∞

1 s

Rs

0 T_tµdt, µ ∈ M(X). However, in general (actually, in most cases of interest), given µ∈ M(X), the map t 7→ Ttµ, t ≥0, is not strongly measurable because the range of the map fails to be separable (for the terminology used here, see Appendix E of Cohn [1]). Fortunately, we can consider a very simple vector integral that we call the pointwise (vector) integral, and that can be used to state and prove a weak* mean ergodic theorem which in turn allows us to prove the following result.

Theorem 1.1 (Pointwise Mean Ergodic Theorem for Transition Func- tions). Let (Pt)t∈[0,+∞) be a Feller transition function that is C0(X)-strongly continuous and C0(X)-equicontinuous (note that, as pointed out earlier, such a transition function satisfies the s.m.a., as well). Then lim

s→∞

1 s

Rs

0 S_tf(x) dt exists and is a real number whenever f ∈C0(X) andx∈X.

Remark. It is easy to find transition functions that satisfy the conditions of the above theorem, and, therefore, for which the conclusion of the theorem holds true. A particularly large and interesting family of examples can be obtained as follows (for details on unexplained terminology, see [7]).

Assume that (X, d) is a locally compact separable metric semigroup with identity u. We use ∗ to denote the convolution operation on M(X). If µ ∈ M(X), then we use the notationµ⁰=δu(δuis the Dirac measure concentrated

(5)

at u; throughout the paper,δx is the Dirac measure concentrated at x ∈X), µ¹ = µ, and µⁿ = µ∗µ∗ · · · ∗µ

| {z }

ntimes

for n ∈ N, n ≥ 2. Let µ ∈ M(X) be a probability measure. It can be shown that the series

∞

P

k=0 α^k

k!µ^k converges in the norm topology of M(X) to a positive element να of M(X) and that kν_αk=e^α (where, of course,e=

∞

P

k=0 1

k!) whenever α∈R,α >0. Setµ₀ =δu, and µ_α =e^−α

∞

P

k=0 α^k

k!µ^k for every α ∈ R, α > 0. Then µ_α, α ∈ [0,+∞) are probability measures, and it can be shown that µ_α+β = µ_α∗ µ_β for every α ∈ [0,+∞) and β ∈ [0,+∞) (the family (µ_α)_{α∈[0,+∞)} is an example of a one-parameter convolution semigroup; for a comprehensive discussion of such semigroups when X is a group, see Chapter III and Chapter IV of Heyer [4], a standard reference to many topics including one-parameter semigroups).

For every α ∈ [0,+∞), let P_α be the transition probability defined by µ_α (see [7] for details). It can be shown that (Pα)α∈[0,+∞) is a Feller transition function. If µis an equicontinuous probability measure, then it can be shown that (P_α)_{α∈[0,+∞)} satisfies all the conditions of Theorem 1.1. Thus, if µ is equicontinuous, the conclusion of the theorem holds true for (Pα)α∈[0,+∞). In particular, the conclusion of Theorem 1.1 holds true for any transition function (P_α)_{α∈[0,+∞)} defined by any probability measureµ whenever (X, d) is a compact metric semigroup.

Our goal in the next section is to define the above-mentioned vector integral and to discuss several of its properties that are useful for proving Theorem 1.1. In order to keep the length of this paper within reasonable limits, we will not prove Theorem 1.1 here. The proof of Theorem 1.1 will be included in the monograph that we mentioned earlier that we are currently writing.

2. THE POINTWISE VECTOR INTEGRAL

Let (Ω,Σ, ν) be a measure space, letY be a nonempty set, and letF be a collection of real-valued functions defined on Y (usuallyF is a Banach lattice of such functions). We say that a function ϕ: Ω→ F ispointwise measurable if the condition below is satisfied.

(PM) For every y ∈ Y, the function ϕy : Ω → R defined by ϕy(t) = ϕ(t)(y) for everyt∈Ω is Σ-measurable.

A pointwise measurable function ϕ : Ω → F is said to be pointwise integrable if the two conditions below are satisfied.

(PI1) For everyy∈Y, the function ϕy, defined at (PM), is integrable.

(6)

(PI2) The function I : Y → R defined by I(y) = R

Ωϕ(t)(y) dν(t) for every y∈Y belongs to F.

If ϕ is pointwise integrable, then the function I defined at (PI2) is called the pointwise integral of ϕ and is denoted by P-R

Ωϕ(t) dν(t). For y ∈ Y, we will use the notation P-R

Ωϕ(t)(y) dν(t) for I(y), rather than P-R

Ωϕ(t) dν(t)

(y), which is the formally correct notation, but it is cumbersome.

RemarkA. We mentioned earlier that we believe strongly that the pointwise integral has appeared often in implicit form in the literature earlier. In support of the above assertion, let us discuss briefly how one can state a re- stricted form of Fubini’s theorem in terms of pointwise integrals. Let (Y,Y, µ1) and (Z,Z, µ₂) be two finite measure spaces, letY ⊗Zbe the productσ-algebra ofY andZ onY×Z, and letµ₁⊗µ₂ be the product measure ofµ₁ andµ₂ on the measurable space (Y ×Z,Y ⊗ Z). Now, let g:Y ×Z →Rbe a bounded Y ⊗ Z-measurable function. For every y ∈Y, let g_y : Z → R be defined by g_y(z) = g(y, z) for every z ∈ Z and, for every z ∈ Z, let g^(z) : Y → R be defined by g^(z)(y) =g(y, z) for every y ∈Y. Finally, let Bb(Y), Bb(Z), and B_b(Y ×Z) be the Banach spaces of all real-valued bounded measurable functions defined on Y, Z, and Y ×Z, respectively. Now let ϕ : Y → Bb(Z) and ψ : Z → B_b(Y) be defined by ϕ(y) = g_y and ψ(z) = g^(z) for every y ∈ Y and z ∈ Z, respectively. It is easy to see that both ϕ and ψ are pointwise measurable, and that both ϕ and ψ satisfy condition (PI1).

By Fubini’s theorem, the pointwise integrals of ϕ and ψ exist and they are:

θϕ : Z → R, θϕ(z) = R

Y ϕ(y)(z) dµ1(y) for every z ∈ Z and θψ : Y → R, θ_ψ(y) =R

Zψ(z)(y) dµ₂(z) for everyy∈Y, respectively. Moreover, using again Fubini’s theorem, we obtain that both θϕ and θψ are µ2- and µ1-integrable, respectively, and

Z

θ_ϕ(z) dµ₂(z) = Z

Y

θ_ψ(y) dµ₁(y) = Z

Y×Z

g(y, z) d(µ₁⊗µ₂)(y, z).

Note that the definitions ofϕ and θ_ϕ, and the measurability of θ_ϕ as a real-valued function defined on the measurable space (Z,Z) do not depend on µ₂. Therefore, by Remark A, the following result is obviously true.

Proposition 2.1. Let (Y,Y, µ) be a finite measure space, let (Z,Z) be a measurable space, let B_b(Z) be the Banach lattice of all real-valued bounded Z-measurable functions defined on Z, and let g : Y ×Z → R be a bounded function measurable with respect to the product σ-algebra Y ⊗ Z. Then the function ϕ : Y → Bb(Z), ϕ(y) = gy for every y ∈ Y, where gy : Z → R, g_y(z) = g(y, z) for every z ∈ Z, is well-defined (in the sense that g_y ∈ Bb(Z) for every y ∈ Y) and is pointwise integrable. The pointwise integral

(7)

P-R

Y ϕ(y) dµ(y) (which is a real-valuedZ-measurable bounded function on Z) is defined by

P- Z

Y

ϕ(y)(z) dµ(y) = Z

Y

g_y(z) dµ(y) = Z

Y

g(y, z) dµ(y) for every z ∈ Z, where (as mentioned earlier) P-R

Y ϕ(y)(z) dµ(y) stands for P-R

Y ϕ(y) dµ(y) (z).

We will call the pointwise integral P-R

Y ϕ(y) dµ(y) that appears in Propo- sition 2.1 a B_b(Z)-pointwise integral. The next result shows that such pointwise integrals appear quite naturally when dealing with transition functions.

Proposition 2.2. Let (Pt)t∈[0,+∞) be a transition function defined on a locally compact separable metric space (X, d), suppose that (Pt)_t∈[0,+∞) satisfies the s.m.a., and let ((St, Tt))_t∈[0,+∞) be the family of Markov pairs defined by (P_t)_t∈[0,+∞). Then, for every f ∈ B_b(X) and every Lebesgue measurable subset L of [0,+∞) of finite Lebesgue measure, the map ϕ^(L)_f : L → B_b(X) defined by ϕ^(L)_f (t) =S_tf for every t∈L is pointwise integrable.

Proof. We apply repeatedly Proposition 2.1 in the case where Y =L,µ is the Lebesgue measure on Y,Z =X, Z =B(X), and g^(f) :L×X →R is defined by g^(f⁾(t, x) =S_tf(x) for every (t, x)∈L×X,f ∈B_b(X).

First, assume that f = 1A for some A ∈ B(X). Then, using (1.1), we obtain that S_t1_A(x) =P_t(x, A) for everyt∈L and x∈X. Since (P_t)_t∈[0,+∞) satisfies the s.m.a., the map (t, x) 7→ P_t(x, A), (t, x) ∈ L×X is jointly measurable with respect to t and x; as the map is also bounded, we can apply Proposition 2.1 in order to infer that the pointwise integral P-R

LS_tf(x) dt exists in this case.

If f ∈ B_b(X) is a simple function; that is, if there exist n ∈ N, n measurable subsets A₁, A₂, . . . , A_n of X, and n real numbers a₁, a₂, . . . , a_n such that f =

n

P

i=1

a_i1_A_i, then the map (t, x) 7→ S_tf(x), (t, x) ∈ L×X, is jointly measurable with respect to t and x because S_tf(x) =

n

P

i=1

a_iP_t(x, A_i) for every (t, x) ∈ L×X, and the maps (t, x) 7→ Pt(x, Ai), (t, x) ∈ L×X, i= 1,2, . . . , nare jointly measurable with respect to t and x. Thus, applying Proposition 2.1 again, we obtain that the pointwise integral P-R

LS_tf(x) dt exists in this case, as well.

Now, if f ∈ B_b(X), f ≥ 0, then there exists a sequence (f_n)n∈N of simple measurable functions that converges uniformly to f, and such that 0 ≤ f_n ≤ f for every n ∈ N. Since S_t, t ≥ 0 are (positive) contractions of Bb(X), the sequence of maps (t, x) 7→ Stfn(x), (t, x) ∈ L×X, n ∈ N,

(8)

converges pointwise on L×X (actually, the sequence converges uniformly) to the map (t, x)7→S_tf(x), (t, x)∈L×X. Accordingly, the map (t, x)7→S_tf(x), (t, x)∈L×X, is jointly measurable intandx, and is bounded. Thus, we can apply again Proposition 2.1 in order to conclude that P-R

LStf(x) dtexists.

Finally, iff ∈B_b(X) is not necessarily a positive element ofB_b(X), then f =f⁺−f⁻, where f⁺ =f ∨0∈ B_b(X), f⁻ = (−f)∨0 ∈B_b(X), f⁺ ≥0, f⁻ ≥0. Using the above discussion, we obtain that both pointwise integrals P-R

LStf⁺(x) dt and P-R

LStf⁻(x) dt exist, and the maps (t, x) 7→ Stf⁺(x) and (t, x) 7→ S_tf⁻(x) are both jointly measurable with respect to t and x, and bounded; hence, the map (t, x) 7→ Stf(x) = Stf⁺(x)−Stf⁻(x) is also bounded and jointly measurable with respect totandx. Using Proposition 2.1, we obtain that the map ϕ^(L)_f :L→B_b(X) defined by ϕ^(L)_f (t) =S_tf for every t∈L is pointwise integrable.

LetJ ⊆Rbe a finite union of bounded intervals ofR, and think ofJ as the measure space defined by theσ-algebra of the Lebesgue measurable subsets of J and the Lebesgue measure on J. Our goal now is to obtain sufficient conditions under which a function ϕ : J → B_b(X) is pointwise integrable, is Bochner integrable in the sense of Appendix E of Cohn [1], and has the property that the pointwise and Bochner integrals are equal.

Ifϕis Bochner integrable in the sense of Cohn [1], we will use the notation B-R

Jϕ(t) dtfor the Bochner integral of ϕon J.

Theorem 2.3. Let ϕ : J → B_b(X) be a continuous bounded function, and assume that the range of ϕ is included in C_b(X). Then the following assertions are true:

(a)ϕ is pointwise integrable;

(b)ϕ is Bochner integrable in the sense of Appendix E of Cohn [1];

(c) the pointwise and the Bochner integrals of ϕare equal.

Proof. (a) Let g :J ×X →R be defined by g(t, x) = ϕ(t)(x) for every (t, x)∈J×X.

We first note thatgis jointly continuous with respect totandx. Indeed, let ((t_n, x_n))n∈N be a convergent sequence of elements of J ×X, set (t, x) =

n→∞lim(tn, xn), and letε∈R,ε >0. Sinceϕis continuous and (tn)n∈Nconverges to t, there exists mε ∈ N such that kϕ(t_n)−ϕ(t)k< ^ε₂ for every n ≥ mε. Since ϕ(t)∈C_b(X) and (x_n)n∈Nconverges to x, there existsn_ε∈Nsuch that

|g(t, x_n)−g(t, x)|< ₂^ε for everyn≥nε. Clearly, we may choosenε such that n_ε≥m_ε. Then, for everyn∈N,n≥n_ε, we have

|g(t_n, x_n)−g(t, x)| ≤ |g(t_n, x_n)−g(t, x_n)|+|g(t, x_n)−g(t, x)| ≤

≤ kϕ(t_n)−ϕ(t)k+|g(t, x_n)−g(t, x)|< ε 2+ ε

2 =ε.

(9)

Since g is jointly continuous with respect to t and x, it also is jointly measurable with respect to t and x. Clearly, g is bounded (because of the boundedness of ϕ); sinceJ has finite Lebesgue measure, we can apply Propo- sition 2.1 in order to conclude that ϕis pointwise integrable.

(b) Since ϕis continuous, ϕ(J) is a separable subset ofB_b(X) (that is, ϕ has a separable range) and ϕ is measurable (i.e., ϕ⁻¹(E) is a Lebesgue measurable subset of J wheneverE is a Borel subset ofB_b(X)). Accordingly, ϕ is strongly measurable. Now, the map t 7→ kϕ(t)k, t ∈ J is continuous (because |kϕ(t)k − kϕ(s)k| ≤ kϕ(t)−ϕ(s)k for every s ∈ J and t ∈ J, and because ϕ is continuous) and bounded (because ϕ is bounded). Since J has finite Lebesgue measure, the mapt7→ kϕ(t)k,t∈J is integrable. Accordingly, ϕ is Bochner integrable in the sense of Appendix E of Cohn [1].

(c) Note that if ψ:J →B_b(X) is of the formψ(t) =

n

P

i=1

1_A_i(t)f_i, t∈J, for somen∈N,nLebesgue measurable subsetsA₁, A₂, . . . , A_n ofJ andnele- ments (real-valued bounded measurable functions defined on X)f1, f2, . . . , fn

ofB_b(X), thenψis obviously strongly measurable (such a functionψis called a strongly measurable simple function) and Bochner integrable in the sense of Cohn (because J has finite Lebesgue measure), and, by definition, the Bochner integral in the sense of Cohn [1] of ψ is B-R

Jψ(t) dt = Pn

i=1

λ(A_i)f_i, where λ is the Lebesgue measure (on J). Also, ψ is pointwise integrable because for every x ∈ X the function t 7→ Pⁿ

i=1

1_A_i(t)f_i(x), t ∈ J, is Lebesgue integrable since t 7→ Pⁿ

i=1

1_A_i(t)f_i(x), t ∈ J, is a simple measurable real- valued map, and J has finite Lebesgue measure; moreover, the pointwise integral of ψ is P-R

Jψ(t) dt (x) =

n

P

i=1

λ(Ai)fi(x) for every x ∈ X; hence, P-R

Jψ(t) dt= B-R

Jψ(t) dt.

Since ϕ is Bochner integrable in the sense of Cohn [1] (as we proved at (b)), using Proposition E2, p. 351 of Cohn [1], we obtain that there exists a sequence (ψ_k)k∈N of strongly measurable simple functions, ψ_k : J → B_b(X) for everyk∈N, such that (ψ_k(t))k∈Nconverges uniformly onX(that is, in the norm topology of Bb(X)) to ϕ(t) for every t∈J, such that kψ_k(t)k ≤ kϕ(t)k for every k ∈ N and t ∈ J, and such that the sequence of Bochner integrals (which do exist as pointed out above) B-R

Jψk(t) dt

k∈Nconverges uniformly on X to B-R

Jϕ(t) dt.

Now, for every x ∈ X, the sequence (u^(x)_k )k∈N, u^(x)_k : J → R, u^(x)_k (t) = ψ_k(t)(x) for every t ∈ J and k ∈ N, converges pointwise to u^(x) : J → R,

(10)

u^(x)(t) = ϕ(t)(x) for every t ∈ J (because (ψk(t))k∈N converges uniformly on X to ϕ(t) for every t ∈ J), so we can apply the dominated convergence theorem to the sequence (u^(x)_k )k∈N in order to infer that the sequence

P-R

Jψ_k(t) dt (x)

k∈N converges to P-R

Jϕ(t) dt

(x). Thus, the sequence P-R

Jψk(t) dt

k∈N converges pointwise onX to P-R

Jϕ(t) dt.

Since (as shown in the first part of the proof) B-R

Jψk(t) dt= P-R

Jψk(t) dt for every k ∈ N, and since B-R

Jψk(t) dt

k∈N converges uniformly on X to B-R

Jϕ(t) dt, we conclude that B-R

Jϕ(t) dt= P-R

Jϕ(t) dt.

It can be shown that the integral discussed in Dunford and Schwartz [3]

is an extension of the classical Bochner integral as presented in Appendix E of Cohn [1]; that is, if ζ :J →B_b(X) is Bochner integrable in the sense of Cohn [1], then ζ is integrable in the sense of Dunford and Schwartz [3] and the two integrals are equal. If we apply Theorem 2.3 to the integrals that appear in Theorem 1.1 and Proposition 2.2, we obtain the following result.

Corollary 2.4. Let (Pt)t∈[0,+∞) be a Feller transition function defined on a locally compact separable metric space (X, d), let ((St, Tt))t∈[0,+∞) be the family of Markov-Feller pairs defined by (Pt)t∈[0,+∞), and assume that (Pt)_t∈[0,+∞)isC0(X)-strongly continuous. IfJ is a finite union of finite length subintervals of [0,+∞) and if f ∈ C₀(X), then the map ϕ^(J)_f : J → B_b(X) defined by ϕ^(J)_f (t) = Stf for every t∈J is pointwise integrable, and Bochner integrable in the sense of Cohn[1]and of Dunford and Schwartz[3]. Moreover, the pointwise integral, and the two Bochner integrals in the sense of Cohn [1]

and in the sense of Dunford and Schwartz [3] are equal.

The proof of the corollary is obvious since the mapϕ^(J)_f that appears in the corollary satisfies the conditions of Theorem 2.3.

The corollary is useful because it allows us to think of integrals of the form R

JS_tfdt,f ∈C₀(X), as Bochner integrals in the sense of Dunford and Schwartz; hence, we can use various results of [3] to study these integrals.

We now turn our attention to another type of pointwise integral that we call the M(X)-pointwise integral.

Let (Ω,Σ, ν) be a measure space, letϕ: Ω→ M(X) be a function, and let us think ofM(X) as a Banach lattice of real-valued functions defined onB(X).

Then it makes sense to consider the definitions stated at the beginning of this section in the present setting. Thus, we say that ϕ is pointwise measurable, or M(X)-pointwise measurable, if, for every A ∈ B(X), the function ψ^(ϕ)_A : Ω → R defined by ψ_A^(ϕ)(t) = ϕ(t)(A) for every t ∈ Ω is measurable, and we say that ϕis pointwise integrable, or M(X)-pointwise integrable if, for every

(11)

A∈ B(X), the function ψ_A^(ϕ) is integrable and the mapI:B(X)→Rdefined by I(A) = R

Ωψ_A^(ϕ)(t) dν(t) for every A ∈ B(X) belongs to M(X). If ϕ is M(X)-pointwise integrable, we callItheM(X)-pointwise integral ofϕand we use the notation P-R

Ωϕ(t) dν(t) rather thanI. Also, it is convenient to use the notation ϕtforϕ(t), t∈Ω, and, for A∈ B(X), to use R

Ωϕt(A) dν(t) for I(A) rather than R

Ωϕ_tdν(t)

(A), which is formally correct, but is cumbersome and less intuitive.

Naturally, the first concern when dealing with an integral is to find conditions for integrability. In the next result, we discuss such conditions. The conditions are general enough for our purposes, and probably they are good enough for most purposes.

Proposition 2.5. Let ϕ : Ω → M(X) be an M(X)-pointwise measurable function. If there exists an integrable function ρ : Ω → R such that

sup

A∈B(X)

|ϕ_t(A)| ≤ ρ(t) for every t ∈ Ω, then ϕ is M(X)-pointwise integrable.

In particular, ϕisM(X)-pointwise integrable whenever the measureν is finite and there exists M ∈Rsuch that|ϕ_t(A)| ≤M for everyt∈ΩandA∈ B(X).

Proof. Letϕ: Ω→ M(X) and ρ: Ω→Rbe as in the statement.

Let A ∈ B(X), and note that the function ψ^(ϕ)_A : Ω → R defined by ψ_A^(ϕ)(t) = ϕ_t(A) for every t ∈ Ω is measurable since ϕ is M(X)-pointwise measurable. Since |ψ^(ϕ)_A (t)| ≤ ρ(t) for every t ∈ Ω and ρ is integrable, ψ^(ϕ)_A is integrable for every A ∈ B(X). Thus, the map I : B(X) → R defined by I(A) = R

Ωψ_A^(ϕ)(t) dν(t) for every A ∈ B(X) is well-defined; accordingly, in order to prove that ϕ is M(X)-pointwise integrable, we have to show that I belongs to M(X). Since I(∅) = 0, in order to prove thatI∈ M(X), we only have to prove that Iis σ-additive.

To this end, let (An)n∈N be a sequence of mutually disjoint measurable subsets of X. We have to prove that

(2.1) I ∪^∞_n=1An

=

∞

X

n=1

I(An);

that is, that the series

∞

P

n=1

I(An) is convergent and the above equality holds true.

Sinceϕt is a signed measure for everyt∈Ω, we have I ∪ⁿ_i=1A_i

= Z

Ω

ϕ_t(∪ⁿ_i=1A_i) dν(t) =

n

X

i=1

Z

Ω

ϕ_t(A_i) dν(t) =

n

X

i=1

I(A_i)

(12)

for every n ∈ N; thus, if lim

n→∞I (∪ⁿ_i=1A_i) exists, then the series

∞

P

n=1

I(A_n) is convergent, and

∞

P

n=1

I(A_n) = lim

n→∞I (∪ⁿ_i=1A_i). This means that in order to prove (2.1) it is enough to prove that lim

n→∞I (∪ⁿ_i=1Ai) exists and

(2.2) lim

n→∞I (∪ⁿ_i=1Ai) = I (∪^∞_n=1An). Now, note that ψ_∪^(ϕ)n

i=1Ai

n∈N is a sequence of integrable functions that converges everywhere on Ω to ψ^(ϕ)_∪∞

n=1An, ψ_∪^(ϕ)n

i=1Ai(t)

≤ ρ(t) for every n ∈ N and t∈ Ω, and

ψ^(ϕ)_∪^∞

n=1An(t)

≤ρ(t) for every t∈ Ω. Thus, we can apply the Lebesgue dominated convergence theorem to the sequence ψ^(ϕ)_∪n

i=1Ai

n∈N and conclude that lim

n→∞

R

Ωψ_∪^(ϕ)n

i=1Ai(t) dν(t) =R

Ωψ_∪^(ϕ)∞

n=1An(t) dν(t).

Thus, (2.2) is true since I (∪ⁿ_i=1A_i) =R

Ωψ_∪^(ϕ)n

i=1Ai(t) dν(t) for every n∈N and I (∪^∞_n=1An) =R

Ωψ^(ϕ)_∪^∞

n=1An(t) dν(t).

We have therefore proved that I∈ M(X), soϕis pointwise integrable.

Ifν is a finite measure, and there exists M ∈R such that |ϕ_t(A)| ≤ M for every A∈ B(X) andt∈Ω, thenϕis pointwise integrable because we can use the above arguments applied to ϕand ρ=M ·1Ω.

The next result consists of an application of Proposition 2.5 to answer a concern raised in Section 1; namely, under fairly general conditions, we can use the corollary to assign a meaning to integrals of the form Rs

0 Ttµdt, where s ∈ [0,+∞), µ ∈ M(X), and (T_t)_t∈[0,+∞) is the semigroup of operators on M(X) generated by a transition probability.

Corollary 2.6. Let (Pt)_t∈[0,+∞) be a transition function defined on (X, d), assume that (Pt)_t∈[0,+∞) satisfies the s.m.a., let ((St, Tt))_t∈[0,+∞) be the family of Markov pairs defined by (P_t)_t∈[0,+∞), and let L be a Lebesgue measurable subset of [0,+∞) that has finite Lebesgue measure. Then, for every µ∈ M(X), the map t7→Ttµ, t∈L, isM(X)-pointwise integrable.

Proof. Letµ∈ M(X), and letϕ:L→ M(X),ϕt=Ttµfor every t∈L.

We have to prove that ϕis M(X)-pointwise integrable.

We first note that ϕ is M(X)-pointwise measurable. Indeed, let A ∈ B(X), and note that, since (P_t)_t∈[0,+∞) satisfies the s.m.a., we can apply Proposition 5.2.1-(a), p. 159 of Cohn [1] to the measure spaces (X,B(X), µ) and (L,L(L), λ), whereL(L) is theσ-algebra of all Lebesgue measurable subsets of L and λ is the Lebesgue measure, and to the map (t, x) 7→ P_t(x, A), (t, x)∈L×X, in order to obtain that the mapt7→ϕt(A) =R

XPt(x, A) dµ(x),

(13)

t ∈L, is measurable; accordingly, the M(X)-pointwise measurability ofϕ is established.

Now, note that |ϕ_t(A)| ≤ |ϕ_t|(A) ≤ |ϕ_t|(X) = |T_tµ|(X) ≤ Tt|µ|(X) = kT_t|µ|k = k|µ|k = kµk for every t ∈ L and A ∈ B(X). Since L has finite Lebesgue measure, using Proposition 2.5 we obtain thatϕisM(X)-pointwise integrable.

In the next result we discuss a useful relationship between the B_b(X)- pointwise integral P-R

LStfdt and the M(X)-pointwise integral P-R

LTtµdt in the case in which we deal with a family ((S_t, T_t))_t∈[0,+∞) of Markov pairs defined by a transition function (P_t)_t∈[0,+∞) that satisfies the s.m.a., f ∈ B_b(X), µ ∈ M(X), and L is a Lebesgue measurable subset of [0,+∞) that has finite Lebesgue measure.

Proposition 2.7. Let (P_t)_t∈[0,+∞) be a transition function defined on (X, d), assume that (P_t)_t∈[0,+∞) satisfies the s.m.a., let ((S_t, T_t))_t∈[0,+∞) be the family of Markov pairs defined by (Pt)t∈[0,+∞), and let L be a Lebesgue measurable subset of [0,+∞) that has finite Lebesgue measure. Then

(2.3)

P-

Z

L

Stfdt, µ

=

f,P- Z

L

Ttµdt

for every f ∈Bb(X) andµ∈ M(X).

Note that (2.3) makes sense because, by Proposition 2.2, the B_b(X)- pointwise integral P-R

LStfdt exists while the M(X)-pointwise integral P- R

LT_tµdtexists by Corollary 2.6.

Proof. We first note that it is easy to see that it is enough to prove the proposition under the assumption that µ≥0. Thus, assume thatµ≥0.

We will prove that (2.3) holds true in three steps: first for f = 1_A for some A ∈ B(X), then for the case when f is a simple B(X)-measurable function and, finally for the general case when f ∈B_b(X) is not necessarily a simple function.

Step 1. Assume that f = 1_A for some A ∈ B(X). The fact that (Pt)t∈[0,+∞)satisfies the s.m.a. allows us to apply Proposition 5.2.1-(b), p. 159 of Cohn [1], and we obtain that

P-

Z

L

St1Adt, µ

= Z

X

P-

Z

L

St1A(x) dt

dµ(x) =

= Z

X

Z

L

P_t(x, A) dt

dµ(x) = Z

L

Z

X

P_t(x, A) dµ(x)

dt=

= P- Z

L

T_tµ(A) dt=

1_A,P- Z

L

T_tµdt

.

(14)

Step 2. If f is a B(X)-measurable simple function (that is, if there exist m ∈ N, m measurable subsets A₁, A₂, . . . , A_m of X and m real numbers a₁, a₂, . . . , a_m such that f =

m

P

i=1

a_i1_A_i), then, using Step 1, we obtain easily that (2.3) holds true forf.

Step3. Assume now thatf ∈Bb(X). Clearly, it is enough to prove that (2.3) is true under the assumption that f ≥0.

Thus, assume that f ≥0. Then there exists a monotone nondecreasing sequence (fn)n∈N of real-valued positive simple B(X)-measurable functions on X such that (f_n)n∈N converges uniformly to f. Therefore, the sequence

fn,P-R

LTtµdt

n∈N converges to f,P-R

LTtµdt .

SinceSt is a positive contraction ofB_b(X), the sequence (Stfn)n∈Ncon- verges uniformly on X for every t. Thus, for every x ∈ X, the sequence of positive functions (ξn^(x))n∈N, ξ^(x)n : L → R, ξ^(x)n (t) = Stfn(x) for every t ∈ L and n∈N, is monotone nondecreasing and converges pointwise on Ltoξ^(x) : L→R,ξ^(x)(t) =Stf(x) for everyt∈L; accordingly, by the monotone convergence theorem, the sequence R

Lξn^(x)(t) dt

n∈N converges toR

Lξ^(x)(t) dt. The fact that R

Lξn^(x)(t) dt

n∈N converges to R

Lξ^(x)(t) dt for every x ∈ X means that the sequence of functions P-R

LStfndt

n∈Nconverges pointwise onX to P-R

LS_tfdt. Since P-R

LS_tf_ndt

n∈N is a monotone nondecreasing sequence of positive B(X)-measurable functions that converges pointwise on X to P- R

LStfdt, using the monotone convergence theorem in the space (X,B(X), µ) we deduce that the sequence R

X P-R

LStfn(x) dt

dµ(x)

n∈N converges to R

X P-R

LStf(x) dt

dµ(x); that is, the sequence hP-R

LStfndt, µi

n∈N converges to

P-R

LS_tfdt, µ . Since by Step 2

P-R

LS_tf_ndt, µ

=

f_n,P-R

LT_tµdt

for every n ∈ N, we conclude that

P-R

LStfdt, µ

= f,P-R

LTtµdt

.

If L is an interval with endpoints a and b, a ≤b, and if ϕ is a B_b(X)- valued function on L, then we denote the B_b(X)-pointwise integral and the Bochner integral of ϕ by P-Rb

aϕtdt and B-Rb

aϕtdt, respectively; similarly, if ϕ is an M(X)-valued function on L, then P-Rb

aϕtdt stands for the M(X)- pointwise integral of ϕ.

In Section 1 we pointed out that, in order to obtain a weak* mean ergodic theorem dealing with the averages ¹_tRt

0 T_sµds, t > 0, where µ ∈ M(X) and (Tt)t∈[0,+∞) is the semigroup of operators defined on M(X) by a transition function, we have to assign a meaning to the integrals Rt

0T_sµds, t ≥ 0. In view of our discussion so far, we define these integrals to be M(X)-pointwise integrals; by Corollary 2.6, the integrals exist whenever the transition function

(15)

defining (Tt)t∈[0,+∞) satisfies the s.m.a. Naturally, the integrals are denoted P-Rt

0T_sµds,t≥0.

As usual, given a functionϕ: (0,+∞)→ M(X), we say thatϕconverges in the weak* topology of M(X) as t → +∞ if there exists µ∈ M(X) such that the limit lim

t→+∞hf, ϕ(t)i exists and is equal tohf, µi for everyf ∈C0(X);

in such a case we also say that the limit of ϕin the weak* topology ofM(X) exists ast→+∞and is equal toµ. We callµthe weak* limit ofϕast→+∞

and we denote µby w*- lim

t→+∞ϕ(t).

Our discussion so far allows us to state the weak* mean ergodic theorem mentioned in Section 1 that can be used to prove the pointwise mean ergodic theorem for transition functions (Theorem 1.1).

Theorem2.8 (Weak* Mean Ergodic Theorem for Transition Functions). Let (P_t)_t∈[0,+∞) be a Feller transition function that is C₀(X)-strongly continuous and C₀(X)-equicontinuous. Then, for every µ ∈ M(X), the limit w*- lim

t→+∞

1 tP-Rt

0T_sµds exists and is an invariant element for (T_t)_t∈[0,+∞) in the sense that if µ^∗ = w*- lim

t→+∞

1 tP-Rt

0Tsµds, then Ttµ^∗ = µ^∗ for every t ∈ [0,+∞); if µ≥0, then µ^∗≥0, as well.

We will not prove the theorem here; a proof will be given in the book that we mentioned earlier that we are currently writing.

Note that if µ = δ_x, x ∈ X and L = [0, s], s ∈ [0,+∞), then (2.3) becomes

(2.4)

Z s

0

S_tf(x) dt=

P- Z s

0

S_tfdt, δ_x

=

f,P- Z s

0

T_tδ_xdt

for every f ∈B_b(X). Using (2.4) we can easily see that, indeed, as mentioned before stating Theorem 2.8, the truth of the theorem implies that Theorem 1.1 holds true. Equation (2.3) of Proposition 2.7 is also used in the proof of Theo- rem 2.8 in order to “transfer” the study ofM(X)-pointwise integrals toBb(X)- pointwise integrals; since under the conditions of Theorem 2.8, Corollary 2.4 can be used, we obtain that theBb(X)-pointwise integrals under consideration are also Bochner integrals, so we can use the powerful machinery of Dunford and Schwartz [3], and then, using (2.3) again, we can “return” to M(X)- pointwise integrals.

Acknowledgements. Many thanks are due to Nicolae Dinculeanu for an exchange of e-mails that has been extremely helpful in writing the paper, and to Jan van Cas- teren for comments on a previous version of the article that have been very useful in improving the work.

(16)

REFERENCES [1] D.L. Cohn,Measure Theory.Birkh¨auser, Boston, 1980.

[2] N. Dinculeanu, Vector Integration and Stochastic Integration in Banach Spaces. Wiley, New York, 2000.

[3] N. Dunford and J.T. Schwartz, Linear Operators, Part I: General Theory. Wiley, New York, 1988.

[4] H. Heyer,Probability Measures on Locally Compact Groups. Springer, Berlin–Heidelberg–

New York, 1977.

[5] H. H. Schaefer,Banach Lattices and Positive Operators. Springer, New York–Heidelberg–

Berlin, 1974.

[6] R. Zaharopol, Invariant Probabilities of Markov-Feller Operators and Their Supports.

Birkh¨auser, Basel, 2005.

[7] R. Zaharopol, Invariant probabilities of convolution operators. Rev. Roumaine Math.

Pures Appl.50(2005), 387–405.

[8] R. Zaharopol,An ergodic decomposition defined by transition probabilities. Submitted.

Received 30 March 2007 Mathematical Reviews

416 Fourth Street Ann Arbor, MI 48103, USA

[email protected]