MULTIDIMENSIONAL AND MULTIVALUED ERGODIC THEOREMS FOR MEASURE PRESERVING TRANSFORMATIONS

The Birkhoff’s Ergodic Theorem (BET) is one of the most important and beautiful results of probability theory. The study of ergodic theorems was started in 1931 by von Neumann and Birkhoff, having its origins in statistical mechanics. In recent decades, the BET has been considerable interest in the multivalued case. In this context, J. Ban 1 obtained the BET for random compact sets and fuzzy random variables in ´ Banach spaces with Hausdorff distance; C. Choirat, C. Hess and R. A. Seri 5 established the BET for normal integrands and consequently for convex random sets in finite dimensional case with respect to Kuratowski convergence; H. Ziat 18 proved the BET for unbounded nonconvex random sets with respect to Mosco convergence, Wijsman convergence and Slice convergence

MULTIDIMENSIONAL AND MULTIVALUED ERGODIC THEOREMS FOR MEASURE - PRESERVING TRANSFORMATIONS Nguyen Van Quang∗, Duong Xuan Giap† Abstract The aim of this paper is to establish some multidimensional and multivalued Birkhoff’s ergodic theorems for measure preserving transformations. Our results generalize and also improve related previously reported results. Mathematics Subject Classifications (2010): 26E25, 28B20, 28D05, 37A05, 37A30, 47A35, 47H04, 49J53, 52A22, 54C60, 58C06, 60G10. Key words and phrases: separable Banach space, random set, measure-preserving transformation, Birkhoff’s ergodic theorem. 1 Introduction The Birkhoff’s Ergodic Theorem (BET) is one of the most important and beautiful results of probability theory. The study of ergodic theorems was started in 1931 by von Neumann and Birkhoff, having its origins in statistical mechanics. In recent decades, the BET has been considerable interest in the multivalued case. In this context, J. B´an [1] obtained the BET for random compact sets and fuzzy random variables in Banach spaces with Hausdorff distance; C. Choirat, C. Hess and R. A. Seri [5] established the BET for normal integrands and consequently for convex random sets in finite dimensional case with respect to Kuratowski convergence; H. Ziat [18] proved the BET for unbounded non-convex random sets with respect to Mosco convergence, Wijsman convergence and Slice convergence. In 1951, N. Dunford introduced the interesting question about the validity of the BET for two-dimensional case raised by H. E. Robbins. As early as 1951 N. Dunford [6] and A. Zygmund [19], as an affirmative answer to this question, obtained some interesting individual ergodic theorems for noncommutative families of measure-preserving transformations with discrete and continuous parameters, respectively. Then later, N. Dunford and J. T. Schwartz [7] and N. A. Fava [8] generalized these results to those at the operator theoretic level. Moreover, many authors generalized them in the direction of weighted averages such as T. Yoshimoto [17], R. L. Jones and J. Olsen [12], M. Lin and M. Weber [14], F. Mukhamedov, M. Mukhamedov and S. Temir [15], etc. However, the results of multi-dimensional BET were only established for the single-valued functions. The aim of this paper is to establish some multidimensional and multivalued Birkhoff’s ergodic theorems for measure preserving transformations. We stress that the usual convexification technique developed in previous studies is no longer applicable because we deal BET with two-dimensional case. To give the main results, we have to use a new method in building structure of double array of selections to prove the “lim inf” part of Mosco convergence. This paper is organized as follows. In Section 2, we introduce some basic notions of random set, fuzzy random set, Mosco convergence, Wijsman convergence and the ergodicity. Section 3 is first concerned with the multi-dimensional single-valued BET for measure-preserving transformations which is more perfect than the result of N. Dunford [6] towards the measurability of limit function. Later, we state the Mosco convergence of two-dimensional multivalued BET for non-convex random sets and fuzzy random integrands in separable Banach space. We also obtain the two-dimensional multivalued BET for Wijsman convergence. ∗ Department of Mathematics, Vinh University, Nghe An Province, Viet Nam. Email: nvquang@hotmail.com † Department of Mathematics, Vinh University, Nghe An Province, Viet Nam. Email: dxgiap@gmail.com 1 2 Preliminaries Throughout this paper, let (Ω, A , P) be a complete probability space, (X, . ) be a separable Banach space and X∗ be its topological dual. In the present paper, R (resp. N) will be denoted the set of all real numbers (resp. positive integers). Let c(X) be the family of all nonempty closed subsets of X. For each A, B ⊂ X, clA, coA denote the norm-closure and the closed convex hull of A, respectively; the distance function d (., A) of A, the norm A of A and the support function s(., A) of A are defined by d (x, A) = inf{ x − y : y ∈ A}, (x ∈ X), A = sup{||x|| : x ∈ A}, ∗ s(x , A) = sup{〈x ∗ , y〉 : y ∈ A}, (x ∗ ∈ X∗ ). Let P (X) be the family of all nonempty subsets of X. In P (X), one defined Minkowski addition and scalar multiplication as follows: A + B = {a + b : a ∈ A, b ∈ B }, λA = {λa : a ∈ A}, where A, B ∈ P (X), λ ∈ R. Let BX be the Borel σ-field on X and Bc(X) be the σ-field on c(X) generated by the sets U − = {C ∈ c(X) : C ∩U = } taken for all open subsets U of X. A mapping F from Ω to c(X) is said to be measurable if F is (A , Bc(X) )-measurable, i.e., for every open set U of X, the subset F −1 (U − ) := {ω ∈ Ω : F (ω) ∩U = } belongs to A . Such a mapping F is called a random set or multivalued (closed-valued) random variable. An F : Ω → c(X) is measurable if and only if there exists a sequence { f n } of random elements f n : Ω → X such that F (ω) = cl{ f n (ω)} for all ω ∈ Ω. Such a sequence { f n } is called a Castaing representation of F . Given the random set F , we define a sub-σ-field AF of A by AF = {F −1 (U ) : U ∈ Bc(X) }, where F −1 (U ) = {ω ∈ Ω : F (ω) ∈ U }, i.e., AF is the smallest sub-σ-field of A with respect to which F is measurable. A random element (Banach space valued random variable) f : Ω → X is called a selection of the random set F if f (ω) ∈ F (ω) for almost all ω ∈ Ω. For every sub-σ-field F of A and for 1 ≤ p < ∞, L p (Ω, F , P, X) denotes the Banach space of 1 (equivalence classes of) F -measurable random elements f : Ω → X such that the norm f p = (E f p ) p is finite. In special case, L p (Ω, A , P, X) (resp. L p (Ω, A , P, R)) is denoted by L p (X) (resp. L p ). For each F -measurable random set F , define the following closed subset of L p (Ω, F , P, X), p S F (F ) = f ∈ L p (Ω, F , P, X) : f (ω) ∈ F (ω), a.s. . p p If F = A then S F (F ) is denoted for shortly by S F . A random set F : Ω → c(X) is called integrable if the set S F1 is nonempty, and it is called integrable bounded if the random variable F is in L 1 . For any sub-σ-field F of A and any F -measurable random set F , the expectation of F over Ω, with respect to F , is defined by E(F, F ) = E f : f ∈ S F1 (F ) , where E f = Ω f d P is the usual Bochner integral of f . Shortly, E(F, A ) is denoted by EF . We note that EF is not always closed. Let Nd = {n = (n 1 , n 2 , ..., n d ) : n i ∈ N, 1 ≤ i ≤ d } be the set of d -dimensional, d ≥ 1, positive integer lattice points. We will keep “≤” for the usual partial ordering on N, i.e., m ≤ n if m i ≤ n i , 1 ≤ i ≤ d . Denote nmin = min{n 1 , n 2 , . . . , n d }. 2 Let t be a topology on X and {A n : n ∈ Nd } be an array in c(X). We put t - lim inf A n = x ∈ X : x = t - lim x n , x n ∈ A n , ∀n ∈ Nd , nmin →∞ nmin →∞ t - lim sup A n = x ∈ X : x = t - lim x k , x k ∈ A nk , ∀k ∈ Nd kmin →∞ nmin →∞ where {A nk : k ∈ Nd } is a sub-array of {A n : n ∈ Nd }. The sets t - lim infnmin →∞ A n and t - lim supnmin →∞ A n are the lower limit and the upper limit of {A n : n ∈ Nd }, relative to topology t . We obviously have t - lim infnmin →∞ A n ⊂ t - lim supnmin →∞ A n . An array A n : n ∈ Nd converges to A, in the sense of Kuratowski, relatively to the topology t , if the two following equalities are satisfied: t - lim supnmin →∞ A n = t - lim infnmin →∞ A n = A. In this case, we shall write A = t - limnmin →∞ A n ; this is true if and only if the next two inclusions hold t - lim sup A n ⊂ A ⊂ t - lim inf A n . nmin →∞ nmin →∞ Let us denote by s (resp. w) the strong (resp. weak) topology of X. It is easily seen that s- lim infnmin →∞ A n ⊂ w- lim supnmin →∞ A n and s- lim infnmin →∞ A n ∈ c(X) unless it is empty. A subset A is said to be the Mosco limit of the array {A n : n ∈ Nd } denoted by M- limnmin →∞ A n = A if w- lim supnmin →∞ A n = s- lim infnmin →∞ A n = A which is true if and only if w- lim sup A n ⊂ A ⊂ s- lim inf A n . nmin →∞ nmin →∞ The Wijsman convergence on c(X) is the pointwise convergence of distance functions. This means that an array A n : n ∈ Nd of nonempty closed subsets of X converges to A ∈ c(X) with respect to Wijsman convergence, denoted by W- limnmin →∞ A n = A as nmin → ∞ if, for every x ∈ X, one has d (x, A) = lim d (x, A n ). nmin →∞ The corresponding definitions of pointwise convergence and almost sure convergence for an array F n : n ∈ Nd of multivalued functions defined on Ω are clear. In fact, in the above definitions, it suffices to replace A n by F n (ω) and A by F (ω) for almost surely ω ∈ Ω. When X is reflexive, the Wijsman topology is in general weaker than the Mosco topology and is equivalent to it when, in addition, the norm of X is Fr´echet differentiable. An other interesting feature of the Wijsman topology TW is that the space (c(X), TW ) metrizable and separable, and that the Borel σ-field of TW is equal to Bc(X) (see [9]). Concerning expectations, conditional expectations, martingales, Mosco convergence and Wijsman convergence of random sets we refer to G. Beer and J. M. Borwein [2], C. Hess [9], F. Hiai and H. Umegaki [11]. In the following, we describe some basic concepts of fuzzy random variables. A fuzzy set in X is a function u : X → [0, 1]. For each fuzzy set u, the α-level set is denoted by L α u = {x ∈ X : u(x) ≥ α} , 0 ≤ α ≤ 1. It is easy to see that, for every α ∈ (0, 1], L α u = ∩β 0}) will be denoted by L α v (resp. L 0+ v). A fuzzy random integrand v : Ω × X → [0, 1] is called integrable if the fuzzy random set ω → v(ω, ·) has expected value. A fuzzy convex random integrand is a fuzzy random integrand v : Ω× X → [0, 1] such that v(ω, .) is convex for each ω ∈ Ω. By upper semicontinuity and fuzzy convexity, for each ω ∈ Ω and for each α ∈ [0, 1], the level set L α v(ω) := {x ∈ X : v(ω, x) ≥ α} is closed convex. An array f n : n ∈ Nd of random elements is called uniformly integrable iff E f n I ( f n >a) → 0 as a → ∞ uniformly in n. Let θ : Ω → Ω be an A -measurable transformation. We said that θ is a measure-preserving transformation or, equivalently, P is said to be θ-invariant measure, if P θ −1 (A) = P(A) for all A ∈ A . The sets A ∈ A that satisfy θ −1 (A) = A are called θ-invariant sets and constitute a sub-σ-field Iθ of A . We say that θ is an ergodic if Iθ is trivial, i.e., whenever A ∈ Iθ then P(A) = 0 or 1. An A -measurable function f is called θ-invariant if f = f ◦ θ. By the remarks in [13, page 5], a function f is θ-invariant iff it is Iθ -measurable. An extended real-valued measurable function f : Ω → R is called θ-finite if P E( f |Iθ ) = ∞ = 0. For notational convenience, the logarithms are to the base 2, for a ∈ R, log(a ∨ 1) will be denoted by log+ a. 3 Main results At first, we establish the multidimensional BET for measure-preserving transformations. It is more perfect than the result of N. Dunford [6] towards the measurability of limit function. We get that the limit function is conditional expectation with respect to the σ-field of invariant sets which was not possible to find it in the literature for multi-dimensional BET case. The following lemma is the key ingredient for proving the main results. Lemma 3.1. Given a positive integer k. Let θ1 , θ2 , . . . , θk be the commutative measure-preserving transformations of the probability space (Ω, A , P). Then for any f belonging to the Zygmund’s class, i.e., E f log+ f k−1 < ∞, the multiple sequence A m1 ,...,mk f := 1 m1 . . . mk m k −1 m 1 −1 ··· i 1 =0 4 i k =0 i i f (θ11 . . . θkk ) converges a.s. to E( f |Iθ ) as min{m 1 , . . . , m k } → ∞, where Iθ = ∩ki=1 Iθi . Moreover, if θs is ergodic for some s belonging to {1, 2, . . . , k}, the limit function is constant a.s., i.e., E( f |Iθ ) = E f a.s. Proof. At first, we will prove this lemma for the real-valued functions case. We have A m1 ,...,mk f (ω) = = 1 m1 . . . mk m k−1 −1 m k −2 m 1 −1 ··· i 1 =0 i k−1 =0 i k =0 mk − 1 1 mk m 1 . . . m k−1 (m k − 1) + = i i f (θ11 . . . θkk θk (ω)) + m k−1 −1 m k −2 m 1 −1 ··· i k−1 =0 i k =0 i 1 =0 m k−1 −1 m 1 −1 ··· i k−1 =0 i 1 =0 i i k−1 (ω)) f (θ11 . . . θk−1 i i f (θ11 . . . θkk θk (ω)) m k−1 −1 m 1 −1 1 1 i k−1 i (ω)) f (θ11 . . . θk−1 ··· m k m 1 . . . m k−1 i 1 =0 i k−1 =0 mk − 1 1 A m1 ,...,mk−1 ,mk −1 f (θk (ω)) + A m1 ,...,mk−1 f (ω). mk mk (3.1) By virtue of Zygmund’s result in [19], there exist the subsets A i , i = 1, 2 with probability one of Ω such that A m1 ,...,mk f (ω) → f (ω) as min{m 1 , . . . , m k } → ∞ for all ω ∈ A 1 (3.2) A m1 ,...,mk−1 f (ω) → f 1 (ω) as min{m 1 , . . . , m k } → ∞ for all ω ∈ A 2 , (3.3) and where both f and f 1 in L 1 . Since θk is a measure-preserving transformation, i.e., P(θk−1 A) = P(A) for all A ∈ A , by choosing A = A 1 and by putting A 3 = θk−1 A 1 , we have P(A 3 ) = 1. Setting B = ∩3i =1 A i , we check that P(B ) = 1. Thus for every ω ∈ B , in (3.1), letting min{m 1 , . . . , m k } → ∞, we obtain f (ω) = f (θk (ω)). It is equivalent to f = f ◦ θk a.s, and hence f is Iθk -measurable. Since the transformations θ1 , θ2 , . . . , θk are commutative, the limit function f is Iθi -measurable for every i = 1, . . . , k. Therefore, we have that f is Iθ -measurable where Iθ = ∩ki=1 Iθi . Let A of Iθ . Since I A is Iθi -measurable for every i = 1, . . . , k, it is θi -invariant which entails E((A m1 ,...,mk f )I A ) = E =E 1 m1 . . . mk 1 m1 . . . mk m k −1 m 1 −1 ··· i 1 =0 i k =0 m k −1 m 1 −1 ··· i 1 =0 i k =0 i i f (θ11 . . . θkk ) I A i i ( f I A )(θ11 . . . θkk ) = E( f I A ). (3.4) By the similar arguments as in proof of [13, Theorem 2.3, page 9], the multiple sequence A m1 ,...,mk ( f I A ) is uniformly integrable and converges a.s. to f I A according to (3.2). It follows from (3.4) that E( f I A ) = E( f I A ). So we conclude that f = E( f |Iθ ). Secondly, in the case of Banach space-valued functions, we proceed as follows: For any x ∈ X and B ∈ A , by using the BET for real-valued function I B , we have that A m1 ,...,mk (x I B ) − E(x I B |Iθ ) = x A m1 ,...,mk I B − E(I B |Iθ ) = x A m1 ,...,mk I B − E(I B |Iθ ) tends to 0 a.s. as min{m 1 , . . . , m k } → ∞. So, the BET is proved to the finitely valued functions case. For any sequence of finitely valued functions { f n : n ≥ 1}, one has A m1 ,...,mk f − E( f |Iθ ) ≤ A m1 ,...,mk ( f − h n ) + A m1 ,...,mk h n − E(h n |Iθ ) + E(h n |Iθ ) − E( f |Iθ ) . (3.5) By the BET to finitely valued functions, the second term on the right side of the inequality (3.5) tends to 0 a.s. for any fixed n as min{m 1 , . . . , m k } → ∞. 5 Consequently, for every n, lim sup min{m 1 ,...,m k }→∞ A m1 ,...,mk f − E( f |Iθ ) ≤ lim sup min{m 1 ,...,m k }→∞ A m1 ,...,mk ( f − h n ) + E(h n |Iθ ) − E( f |Iθ ) . (3.6) By the choice of {h n : n ≥ 1}, the last terms on the right side of (3.6) tends to 0 as n → ∞. Finally, since the triangle inequality for the norm and the identity g ◦ θ i = g ◦ θ i , one has A m1 ,...,mk ( f − h n ) ≤ A m1 ,...,mk f − h n . By the BET for real-valued functions case A m1 ,...,mk f − h n tends a.s. to a limit function with integral E f − h n . This can again be made arbitrarily small. Therefore, we obtain the desired conclusion. In the case of extended real-valued random variables, we obtain the following result. Theorem 3.2. Let θ1 , θ2 be two measure-preserving transformations of the probability space (Ω, A , P). Then for every quasi-integrable extended real-valued random variable f satisfying θ1 -finite, one has that (a) 1 m−1 n−1 j f (θ1i θ2 ) and min{m,n}→∞ mn i =0 j =0 lim sup 1 m−1 n−1 j f (θ1i θ2 ) are θ2 -measurable, min{m,n}→∞ mn i =0 j =0 lim inf (b) if θ1 , θ2 are commutative and if let Iθ = ∩2i =1 Iθi , then 1 m−1 n−1 j f (θ1i θ2 ) ≥ E( f |Iθ ) a.s. min{m,n}→∞ mn i =0 j =0 lim inf (where both sides can be equal to + ∞ or − ∞). Proof. Let A mn f (ω) be as in the proof of Lemma 3.1 with k = 2. Since f is quasi-integrable, it suffices to consider the non-integrable part of f , say the positive one. Therefore, we can restrict our analysis to a positive random variable f . For any m, n ≥ 1 and ω ∈ Ω, we have A mn f (ω) = 1 n −1 A m,n−1 f (θ2 (ω)) + A m f (ω). n n (3.7) By using [10, Theorem 1], we get lim A m f (ω) = E( f |Iθ1 ) a.s. m→∞ Combining this with the θ1 -finiteness of f and by taking the lim sup as min{m, n} → ∞ on both sides of (3.7), it follows that u(ω) = u ◦ θ2 (ω) a.s. where u(ω) = lim supmin{m,n}→∞ A mn f (ω). This shows the Iθ2 -measurability of u. Similar proofs hold for the inferior limit. Hence, the statement (a) is proved completely. For every integer p ≥ 1, consider the random variable f p defined by f p (ω) = f (ω) 0 if f (ω) ≤ p, otherwise. Since f p is in Zygmund’s class, by applying Theorem 3.1, we have that for every p ≥ 1, lim inf min{m,n}→∞ A mn f (ω) ≥ lim inf min{m,n}→∞ A mn f p (ω) = E( f p |Iθ )(ω) a.s. Letting p → ∞ and invoking the Monotone Convergence Theorem for conditional expectation we get the conclusion (b). The crucial step for showing the multivalued Birkhoff’s ergodic theorem for non-convex case with respect to Mosco convergence consists of the following proposition. To give this proposition, we extend the convexification technique to double array case. This result is given under an assumption of ergodicity of the measure-preserving transformation. 6 Proposition 3.3. Let F be an integrable random set with values in c(X) satisfying E( F log+ F ) < ∞ and let θ1 , θ2 be two commutative measure-preserving transformations of (Ω, A , P) such that, for every i ∈ {1, 2} and every integer s ≥ 1, θis is ergodic. Then co EF ⊂ s- m n 1 j F (θ1i θ2 (ω)) a.s. cl min{m,n}→∞ mn i =1 j =1 lim inf j 1 n i Proof. Let G mn (ω) = mn cl m i =1 j =1 F (θ1 θ2 (ω)), m ≥ 1, n ≥ 1, ω ∈ Ω. To prove this proposition, we will use [3, Proposition 3.5]. The proof will be performed in several steps. Step 1. Let x ∈ co EF and ε > 0, by virtue of [3, Lemma 3.6], there exists f 1 , . . . , f k ∈ S F1 such that 1 k k i =1 E( f i ) − x < ε. At first, we will show that 1 k E( f i ) ∈ s- lim inf G mn (ω) a.s. min{m,n}→∞ k i =1 (3.8) is enough to prove the “lim inf” part co EF ⊂ s- lim infmin{m,n}→∞ G mn (ω) a.s. of Mosco convergence. Indeed, let z k = k1 ki=1 E( f i ). Since X is separable, there exists a countable dense set D co EF of co EF . For each fixed x ( j ) ∈ D co EF and for every εk = k1 (k ≥ 1), by (3.8), there exists an element z k of X, which depends on x ( j ) and εk , such that z k ∈ s- lim infmin{m,n}→∞ G mn (ω) a.s. Therefore, for each k ≥ 1, there exists a negligible set Nk ∈ A such that z k ∈ s- lim infmin{m,n}→∞ G mn (ω) for all ω ∈ Ω\Nk . Letting N= ∞ N , then P(N ) = 0. For each ω ∈ N , it follows from the set s- lim infmin{m,n}→∞ G mn (ω) is closed, k=1 k z k ∈ s- lim infmin{m,n}→∞ G mn (ω) for all k and z k → x ( j ) as k → ∞, that x ( j ) ∈ s- lim infmin{m,n}→∞ G mn (ω). This means that x ( j ) ∈ s- lim infmin{m,n}→∞ G mn (ω) a.s., for each fixed j ≥ 1. Noting that D co EF is a countable set, we obtain D co EF ⊂ s- lim infmin{m,n}→∞ G mn (ω) a.s. Since the set s- lim infmin{m,n}→∞ G mn (ω) is closed for each ω, by taking the closure of both sides of the above relation, we have coEF ⊂ s- lim infmin{m,n}→∞ G mn (ω) a.s. Therefore, the above statement is proved. Fixed k, as in [3, 4], we define f i j ∈ S F1 , i , j = 1, . . . , k as follows: f i + j −1 f i + j −1−k fi j = if i + j ≤ k + 1, if i + j > k + 1. It is easy to check that 1 k 1 k k E fi = 2 E fi j . k i =1 k i =1 j =1 (3.9) Next, we define the double array { f i j : i ≥ 1, j ≥ 1} by (t −1)k+ j f (s−1)k+i ,(t −1)k+ j (ω) = f i j (θ1(s−1)k+i θ2 (ω)), i , j ∈ {1, . . . , k}, s, t ≥ 1, ω ∈ Ω. For any m ≥ 1, n ≥ 1, there exist the integers s m , p m , t n and q n satisfying m = s m k + p m , s m ≥ 0, 1 ≤ p m ≤ k, (3.10) n = t n k + q n , t n ≥ 0, 1 ≤ q n ≤ k. (3.11) From the above relationships, we deduce that the sequences (p m ) and (q n ) are bounded, whereas from (3.10) and (3.11) we deduce lim s m = ∞ and lim t n = ∞. m→∞ n→∞ 7 Furthermore, it is not difficult to show that for all ω ∈ Ω, the following equality holds 1 m n sm tn f i j (ω) = mn i =1 j =1 mn + k sm tn k 1 s i =1 j =1 m t n tn pm k 1 mn i =1 j =1 t n l =1 r =1 f (l −1)k+i ,(r −1)k+ j (ω) tn r =1 f sm k+i ,(r −1)k+ j (ω) + + s m k qn 1 mn i =1 j =1 s m sm f (l −1)k+i ,tn k+ j (ω) l =1 1 p m qn f s k+i ,tn k+ j (ω). mn i =1 j =1 m (3.12) The proof will be performed as follows. Step 2. Claim 1: k sm tn mn k 1 s tn m i =1 j =1 sm tn l =1 r =1 f (l −1)k+i ,(r −1)k+ j (ω) → 1 k k E f i j a.s. as min{m, n} → ∞. k 2 i =1 j =1 (3.13) For every integer s ≥ 1, f ∈ S F1 and measure-preserving transformation θ, in view of f ◦ θ s log+ f ◦ θ s E ◦ θs = E f log+ f =E f log+ f ≤ E F log+ F < ∞, the function f ◦ θ s belongs to the Zygmund’s class. For each i , j = 1, . . . , k, according to Lemma 3.1 for two measure-preserving transformations θ1k , θ2k and the function f i j ◦ θ1i belonging to the Zygmund’s class, we have 1 sm tn sm tn l =1 r =1 f (l −1)k+i ,(r −1)k+ j (ω) = 1 sm tn = 1 sm tn sm tn l =1 r =1 sm tn l =1 r =1 (r −1)k+ j f i j θ1(l −1)k+i θ2 (ω) j f i j ◦ θ1i (θ1k )l −1 (θ2k )r −1 ◦ θ2 (ω) j → E f i j ◦ θ1i Iθk ◦ θ2 (ω) a.s. as min{m, n} → ∞ where Iθk = ∩2i =1 Iθk i =E f i j ◦ θ1i j ◦ θ2 (ω) a.s. as min{m, n} → ∞ (by θik is ergodic for some i ) = E( f i j ) a.s. as min{m, n} → ∞. (3.14) We have lim m→∞ 1 1 pm 1 1 1 qn 1 tn sm = lim ( + − ) = and lim = lim ( + − )= . n→∞ n n→∞ k m m→∞ k m mk k n nk k (3.15) Therefore, (3.13) holds. Claim 2: tn pm k 1 mn i =1 j =1 t n tn r =1 f sm k+i ,(r −1)k+ j (ω) + s m k qn 1 mn i =1 j =1 s m sm f (l −1)k+i ,tn k+ j (ω) → 0 a.s. as min{m, n} → ∞. (3.16) l =1 For each m ≥ 1 and i , j ∈ {1, . . . , k}, by applying Lemma 3.1 for the measure-preserving transformation θ2k s k+i and the function f i j ◦ θ1m 1 tn in L 1 (X), we get tn r =1 f sm k+i ,(r −1)k+ j (ω) = 1 tn = 1 tn tn r =1 tn r =1 s k+i (r −1)k+ j θ2 f i j θ1m s k+i f i j ◦ θ1m s k+i → E f i j ◦ θ1m (ω) j (θ2k )r −1 ◦ θ2 (ω) j Iθk ◦ θ2 (ω) a.s. as n → ∞ (by Lemma 3.1) 2 s k+i = E f i j ◦ θ1m j ◦ θ2 (ω) a.s. as n → ∞ (by θ2k is ergodic) = E( f i j ) a.s. as n → ∞. 8 (3.17) Thus, we have that tn pm k 1 mn i =1 j =1 t n tn r =1 f sm k+i ,(r −1)k+ j (ω) → 0 a.s. as min{m, n} → ∞. Similarly, we obtain s m k qn 1 mn i =1 j =1 s m sm f (l −1)k+i ,tn k+ j (ω) → 0 a.s. as min{m, n} → ∞. l =1 Hence, (3.16) is proved. Claim 3: 1 p m qn f s k+i ,tn k+ j (ω) → 0 a.s. as min{m, n} → ∞. mn i =1 j =1 m (3.18) From (3.14), (3.15) and (3.17), we deduce that for each i , j ∈ {1, . . . , k}, sm tn 1 1 f s k+i ,tn k+ j (ω) = mn m mn s m t n sm tn f l k+i ,r k+ j (ω) − l =1 r =1 − → sm − 1 1 . s m (s m − 1)t n s m −1 t n f l k+i ,r k+ j (ω) l =1 r =1 t n − 1 1 tn −1 . f s k+i ,r k+ j (ω) s m t n t n − 1 r =1 m 1 (E f i j − 1.E f i j − 0.E f i j ) = 0 a.s. as min{m, n} → ∞, k2 whence Claim 3 follows by applying this estimate. Final Step and Conclusion: Combining the above limits and using (3.9) and coming back to (3.12) we obtain that 1 m n 1 k f i j (ω) → E f i a.s. as min{m, n} → ∞. mn i =1 j =1 k i =1 Note that from the inclusion 1 m n 1 m n j f i j (ω) ∈ F (θ1i θ2 (ω)) a.s., mn i =1 j =1 mn i =1 j =1 it follows from (3.8) that m n 1 k 1 j E f i ∈ s- lim inf cl F (θ1i θ2 (ω)) a.s. min{m,n}→∞ mn i =1 j =1 k i =1 The proof is therefore completed. Without assuming ergodicity on the measure-preserving transformations, we have the following theorem. Theorem 3.4. Given a positive integer k. Let θ1 , θ2 , . . . , θk be the commutative measure-preserving transformations of the probability space (Ω, A , P). Then for any integrable random set F with values in c(X) satisfying E F log+ F k−1 E(F |Iθ ) ⊂ s- < ∞, we have m k −1 m 1 −1 1 i i cl ··· F (θ11 . . . θkk (ω)) a.s., min{m 1 ,...,m k }→∞ m 1 . . . m k i 1 =0 i k =0 lim inf where Iθ = ∩ki=1 Iθi . 9 Proof. Let A m1 ,...,mk F (ω) = 1 m 1 ...m k cl i m k −1 i F (θ11 . . . θkk (ω)), i k =0 m 1 −1 ··· i 1 =0 ω ∈ Ω. By virtue of [18, Lemma 2.2], we can choose { f n : n ≥ 1} as a sequence in S F1 such that E( f n |Iθ ) : n ≥ 1 be a Castaing representation of E(F |Iθ ) where Iθ = ∩ki=1 Iθi . According to Lemma 3.1 for the functions f n , n ≥ 1 belonging to Zygmund’s class, we get lim min{m 1 ,...,m k }→∞ A m1 ,...,mk f n (ω) = E( f n |Iθ )(ω) a.s. It implies that there exists a negligible set N ∈ A such that for every ω ∈ Ω\N , E( f n |Iθ )(ω) ∈ s- lim inf min{m 1 ,...,m k }→∞ A m1 ,...,mk F (ω) for all integer n ≥ 1. Since the E( f n |Iθ )(ω), n ≥ 1 are dense in E(F |Iθ )(ω) and s- lim infmin{m1 ,...,mk }→∞ A m1 ,...,mk F (ω) is closed, we obtain E(F |Iθ ) ⊂ s- lim inf min{m 1 ,...,m k }→∞ A m1 ,...,mk F (ω) a.s. Next, we prove the “lim sup” part of Mosco convergence of two-dimensional multivalued Birkhoff’s ergodic theorem. Proposition 3.5. Let F be an integrable random set with values in c(X) satisfying E( F log+ F ) < ∞ and let θ1 , θ2 be two commutative measure-preserving transformations of (Ω, A , P) such that θi is ergodic for some i ∈ {1, 2}. Then m n 1 j cl F (θ1i θ2 (ω)) ⊂ co EF a.s. min{m,n}→∞ mn i =1 j =1 w- lim sup Proof. Put G mn (ω) = 1 mn cl m i =1 n i j j =1 F (θ1 θ2 (ω)), m, n ≥ 1, ω ∈ Ω. Let {x k : k ≥ 1} be a dense sequence of X\co EF . Since X is separable, by the separation theorem, there exists a sequence x k∗ : k ≥ 1 in X∗ with x k∗ = 1 such that 〈x k∗ , x k 〉 − d x k , co EF ≥ s x k∗ , co EF , ∀k ≥ 1. (3.19) Thus, x ∈ coEF if and only if 〈x k∗ , x〉 ≤ s x k∗ , co EF for all k ≥ 1. Moreover, it follows from the inequality (3.19) that for every integer k ≥ 1, E s x k∗ , F (·) = s x k∗ , co EF < ∞. Therefore, the real-valued function s(x ∗j , F (·)) is integrable. On the other hand, E s x k∗ , F (·) log+ s x k∗ , F (·) ≤ E F log+ F < ∞ for every k ≥ 1. (3.20) So, according to Lemma 3.1 for real-valued case, there exists a negligible subset N of A such that for every ω ∈ Ω\N and k ≥ 1, s x k∗ ,G mn (ω) = 1 m n j s x k∗ , F θ1i θ2 (ω) mn i =1 j =1 → s x k∗ , co EF < ∞ as min{m, n} → ∞. If y ∈ w- lim supmin{m,n}→∞ G mn (ω) for ω ∈ Ω\N , then there exists y r s ∈ G mr n s (ω) such that 〈x k∗ , y〉 = lim 〈x k∗ , y r s 〉 ≤ min{r,s}→∞ lim min{r,s}→∞ s x k∗ ,G mr n s (ω) = s x k∗ , co EF for all k ≥ 1, which implies y ∈ co EF . Thus w- lim supmin{m,n}→∞ G mn (ω) ⊂ co EF a.s. By applying Propositions 3.3 and 3.5 we obtain immediately the two-dimensional multivalued Birkhoff’s ergodic theorem with respect to the Mosco convergence. 10 Theorem 3.6. Let F be an integrable random set with values in c(X) satisfying E F log+ F < ∞ and let θ1 , θ2 be two commutative measure-preserving transformations of (Ω, A , P) such that, for every i ∈ {1, 2} and every integer s ≥ 1, θis is ergodic. Then M- m n 1 j F θ1i θ2 (ω) = co EF a.s. cl min{m,n}→∞ mn i =1 j =1 lim Now, we extend the above theorem to fuzzy random integrands. Theorem 3.7. Assume that v : Ω ⊗ X → [0, 1] is an integrable fuzzy random integrand such that S L1 1 v = and E L 0+ v log+ L 0+ v < ∞. Let θ1 , θ2 be two commutative measure-preserving transformations of (Ω, A , P) such that, for every i ∈ {1, 2} and every integer s ≥ 1, θis is ergodic. Then M- 1 m n j v θ1i θ2 (ω), · = co Ev(ω, ·) a.s., min{m,n}→∞ mn i =1 j =1 lim that is, there exists a negligible set N ∈ A verifying: M- lim min{m,n}→∞ 1 m n j v θ1i θ2 (ω), · mn i =1 j =1 Lα = L α co Ev(ω, ·) for all α ∈ [0, 1]. j 1 m n i Proof. Let G˜ mn (ω) = mn For each α ∈ [0, 1], L α v(ω) = {x ∈ X : v(ω, x) ≥ α} is i =1 j =1 v(θ1 θ2 (ω), ·). measurable because its graph is measurable. Therefore, by using Theorem 3.6, we have that M- lim min{m,n}→∞ L αG˜ mn (ω) = M- m n 1 j cl L α v(θ1i θ2 (ω), ·) min{m,n}→∞ mn i =1 j =1 lim = co E (L α v(ω)) a.s. = L α co Ev(ω, ·) a.s. for every fixed α ∈ (0, 1], in particular, for every α = r ∈ Q, where Q is the set of all rational numbers. Since countable set Q is dense in [0, 1] and L αG˜ mn (ω) = limr ↑α,r ∈Q L r G˜ mn (ω), there exists a negligible subset N of A verifying M- limmin{m,n}→∞ L αG˜ mn (ω) = L α co Ev(ω, ·) , for every α ∈ (0, 1]. Hence the theorem is completely proved. In the following, we establish the two-dimensional multivalued BET for non-convex random sets with respect to the Wijsman convergence. Theorem 3.8. Let F be an integrable random set with values in c(X) satisfying E F log+ F < ∞ and let θ1 , θ2 be two commutative measure-preserving transformations of (Ω, A , P) such that, for every i ∈ {1, 2} and every integer s ≥ 1, θis is ergodic. Then W- m n 1 j cl F (θ1i θ2 (ω)) = co EF a.s. min{m,n}→∞ mn i =1 j =1 1 Proof. Let G mn (ω) = mn cl m i =1 Step 1. We first show that lim inf lim n i j j =1 F (θ1 θ2 (ω)) for every m, n min{m,n}→∞ ≥ 1 and ω ∈ Ω. d (x,G mn (ω)) ≥ d x, co EF for every x ∈ X, a.s. (3.21) Indeed, the closed unit ball of X∗ is denoted by B ∗ . It is known that for any closed convex subset A of X and for any x ∈ X, we have d (x, A) = sup 〈x ∗ , x〉 − s(x ∗ , A) . x ∗ ∈B ∗ 11 (3.22) By using [9, Lemma 3.1], there exists a countable subset D ∗ of B ∗ such that, for any x ∈ X, d x, co EF = sup 〈x ∗ , x〉 − s(x ∗ , co EF ) . (3.23) x ∗ ∈D ∗ By virtue of Lemma 3.1 for integrable real-valued functions in Zygmund’s class applied to the function s(x ∗ , F (·)) for each x ∗ ∈ D ∗ , we have that for every x ∈ X, lim inf min{m,n}→∞ d (x,G mn (ω)) ≥ = lim inf min{m,n}→∞ lim inf d x, coG mn (ω) sup 〈x ∗ , x〉 − s x ∗ ,G mn (ω) min{m,n}→∞ x ∗ ∈B ∗ ≥ sup x ∗ ∈B ∗ ≥ sup x ∗ ∈D ∗ (by the equality (3.22)) lim inf 〈x ∗ , x〉 − 1 m n j s x ∗ , F θ1i θ2 (ω) mn i =1 j =1 lim inf 〈x ∗ , x〉 − 1 m n j s x ∗ , F θ1i θ2 (ω) mn i =1 j =1 min{m,n}→∞ min{m,n}→∞ = sup 〈x ∗ , x〉 − Es x ∗ , F (·) a.s. (by Lemma 3.1) x ∗ ∈D ∗ = sup 〈x ∗ , x〉 − s(x ∗ , co EF ) a.s. x ∗ ∈D ∗ = d (x, co EF ) a.s. (by the equality (3.23)). Since the set having probability one in above statement doesn’t depend on x, we obtain (3.21). Step 2. To finish the proof, we will show that lim sup d (x,G mn (ω)) ≤ d (x, co EF ) for every x ∈ X, a.s. (3.24) min{m,n}→∞ From the separability of X, there is a countable dense subset D (resp. D ) of X (resp. co EF ). Since the distance function d (x, co EF ) is Lipschitz of x (with Lipschitz constant 1), we only prove (3.24) for all x ∈ D. So, consider x ∈ D and an integer p ≥ 1. One can find y = y(x, p) ∈ co EF , depending on x and p, such that x − y ≤ d (x, co EF ) + 1 . 2p Therefore, there exists y = y (x, p) ∈ D such that x−y ≤ x−y + y−y ≤ x−y + 1 1 ≤ d (x, co EF ) + . 2p p By virtue of Proposition 3.3, there exist a negligible subset N (x, p) of A and a double array { f i j : i ≥ 1, j ≥ 1} of f i j ∈ S 1 i j such that F (θ1 θ2 ) y = 1 m n f i j (ω) for every ω ∈ Ω \ N (x, p). min{m,n}→∞ mn i =1 j =1 lim Define the negligible subset N of A as the union of the N (x, p) where x ∈ D and p ≥ 1. Thus, for any ω ∈ Ω\N and any x ∈ D, lim sup d (x,G mn (ω)) ≤ min{m,n}→∞ lim min{m,n}→∞ x− 1 m n f i j (ω) = x − y mn i =1 j =1 ≤ d (x, co EF ) + 1 , p whence, by the arbitrariness of p, the desired conclusion follows. Acknowledgments This work was completed while the authors were visiting the Vietnam Institute for Advanced Study in Mathematics (VIASM). The authors would like to thank the VIASM for supporting the visit and hospitality. This research was funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED). 12 References [1] J. B´an (1991). Ergodic theorems for random compact sets and fuzzy variables in Banach spaces. Fuzzy Sets and Systems 44, 71-82. [2] G. Beer and J. M. Borwein (1990). Mosco Convergence and Reflexivity. Proceedings of the American Mathematical Society 109 (2), 427-436. [3] C. Castaing, N. V. Quang and D. X. Giap (2012). Mosco convergence of strong law of large numbers for double array of closed valued random variables in Banach space. Journal of Nonlinear and Convex Analysis 13 (4), 615-636. [4] C. Castaing, N. V. Quang and D. X. Giap (2012). Various convergence results in strong law of large numbers for double array of random sets in Banach spaces. Journal of Nonlinear and Convex Analysis 13 (1), 1-30. [5] C. Choirat, C. Hess and R. A. Seri (2003). A functional version of the Birkhoff ergodic theorem for a normal integrand: A variational approach. The Annals of Probability 31 (1), 63-92. [6] N. Dunford (1951). An individual ergodic theorem for non-commutative transformations. Acta Scientiarum Mathematicarum (Szeged) 14, 1-4. [7] N. Dunford and J. T. Schwartz (1956). Convergence almost everywhere of operator averages. J. Rational Mech. Anal. 5, 129-178. [8] N. A. Fava (1972). Weak type inequalities for product operators. Studia Mathematica 42, 271-288. [9] C. Hess (1999). The distribution of unbounded random sets and the multivalued strong law of large numbers in nonreflexive Banach spaces. Journal of Convex Analysis 6 (1), 163-182. [10] C. Hess, R. Seri and C. Choirat (2010). Ergodic theorems for extended real-valued random variables. Stochastic Processes and their Applications 120, 1908-1919. [11] F. Hiai and H. Umegaki (1977). Integrals, conditional expectations and martingales of multivalued functions. Journal of Multivariate Analysis 7, 149-182. [12] R. L. Jones and J. Olsen (1994). Multiparameter weighted ergodic theorems. Canadian Journal of Mathematics 46 (2), 343-356. [13] U. Krengel (1985). Ergodic theorems. Walter de Gruyter Studies in Mathematics 6, Berlin-New York. [14] M. Lin and M. Weber (2007). Weighted ergodic theorems and strong laws of large numbers. Ergodic Theory and Dynamical Systems 27, 511-543. [15] F. Mukhamedov, M. Mukhamedov and S. Temir (2008). On multiparameter weighted ergodic theorem for noncommutative L p -spaces. Journal of Mathematical Analysis and Applications 343, 226-232. [16] M. L. Puri and D. A. Ralescu (1986). Fuzzy random variables. Journal of Mathematical Analysis and Applications 114 (2), 409-422. [17] T. Yoshimoto (1976). An ergodic theorem for noncommutative operators. Proceedings of the American Mathematical Society 54, 125-129. [18] H. Ziat (2011). Ergodic theorem and strong law of large numbers for unbounded and nonconvex random sets. International Journal of Mathematical Analysis 5 (48), 2375-2393. [19] A. Zygmund (1951). An individual ergodic theorem for non-commutative transformations. Acta Scientiarum Mathematicarum (Szeged) 14, 103-110. 13 [...]... Institute for Advanced Study in Mathematics (VIASM) The authors would like to thank the VIASM for supporting the visit and hospitality This research was funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) 12 References [1] J B´an (1991) Ergodic theorems for random compact sets and fuzzy variables in Banach spaces Fuzzy Sets and Systems 44, 71-82 [2] G Beer and J... law of large numbers for double array of random sets in Banach spaces Journal of Nonlinear and Convex Analysis 13 (1), 1-30 [5] C Choirat, C Hess and R A Seri (2003) A functional version of the Birkhoff ergodic theorem for a normal integrand: A variational approach The Annals of Probability 31 (1), 63-92 [6] N Dunford (1951) An individual ergodic theorem for non-commutative transformations Acta Scientiarum... 163-182 [10] C Hess, R Seri and C Choirat (2010) Ergodic theorems for extended real-valued random variables Stochastic Processes and their Applications 120, 1908-1919 [11] F Hiai and H Umegaki (1977) Integrals, conditional expectations and martingales of multivalued functions Journal of Multivariate Analysis 7, 149-182 [12] R L Jones and J Olsen (1994) Multiparameter weighted ergodic theorems Canadian Journal... Krengel (1985) Ergodic theorems Walter de Gruyter Studies in Mathematics 6, Berlin-New York [14] M Lin and M Weber (2007) Weighted ergodic theorems and strong laws of large numbers Ergodic Theory and Dynamical Systems 27, 511-543 [15] F Mukhamedov, M Mukhamedov and S Temir (2008) On multiparameter weighted ergodic theorem for noncommutative L p -spaces Journal of Mathematical Analysis and Applications...Theorem 3.6 Let F be an integrable random set with values in c(X) satisfying E F log+ F < ∞ and let θ1 , θ2 be two commutative measure- preserving transformations of (Ω, A , P) such that, for every i ∈ {1, 2} and every integer s ≥ 1, θis is ergodic Then M- m n 1 j F θ1i θ2 (ω) = co EF a.s cl min{m,n}→∞ mn i =1 j =1 lim Now, we extend the above theorem to fuzzy random integrands Theorem 3.7 Assume that v... theorem to fuzzy random integrands Theorem 3.7 Assume that v : Ω ⊗ X → [0, 1] is an integrable fuzzy random integrand such that S L1 1 v = and E L 0+ v log+ L 0+ v < ∞ Let θ1 , θ2 be two commutative measure- preserving transformations of (Ω, A , P) such that, for every i ∈ {1, 2} and every integer s ≥ 1, θis is ergodic Then M- 1 m n j v θ1i θ2 (ω), · = co Ev(ω, ·) a.s., min{m,n}→∞ mn i =1 j =1 lim that is,... integrable random set with values in c(X) satisfying E F log+ F < ∞ and let θ1 , θ2 be two commutative measure- preserving transformations of (Ω, A , P) such that, for every i ∈ {1, 2} and every integer s ≥ 1, θis is ergodic Then W- m n 1 j cl F (θ1i θ2 (ω)) = co EF a.s min{m,n}→∞ mn i =1 j =1 1 Proof Let G mn (ω) = mn cl m i =1 Step 1 We first show that lim inf lim n i j j =1 F (θ1 θ2 (ω)) for every... Applications 343, 226-232 [16] M L Puri and D A Ralescu (1986) Fuzzy random variables Journal of Mathematical Analysis and Applications 114 (2), 409-422 [17] T Yoshimoto (1976) An ergodic theorem for noncommutative operators Proceedings of the American Mathematical Society 54, 125-129 [18] H Ziat (2011) Ergodic theorem and strong law of large numbers for unbounded and nonconvex random sets International Journal... transformations Acta Scientiarum Mathematicarum (Szeged) 14, 1-4 [7] N Dunford and J T Schwartz (1956) Convergence almost everywhere of operator averages J Rational Mech Anal 5, 129-178 [8] N A Fava (1972) Weak type inequalities for product operators Studia Mathematica 42, 271-288 [9] C Hess (1999) The distribution of unbounded random sets and the multivalued strong law of large numbers in nonreflexive Banach... particular, for every α = r ∈ Q, where Q is the set of all rational numbers Since countable set Q is dense in [0, 1] and L αG˜ mn (ω) = limr ↑α,r ∈Q L r G˜ mn (ω), there exists a negligible subset N of A verifying M- limmin{m,n}→∞ L αG˜ mn (ω) = L α co Ev(ω, ·) , for every α ∈ (0, 1] Hence the theorem is completely proved In the following, we establish the two-dimensional multivalued BET for non-convex random ... fuzzy random integrand such that S L1 v = and E L 0+ v log+ L 0+ v < ∞ Let θ1 , θ2 be two commutative measure- preserving transformations of (Ω, A , P) such that, for every i ∈ {1, 2} and every... integrable random set with values in c(X) satisfying E F log+ F < ∞ and let θ1 , θ2 be two commutative measure- preserving transformations of (Ω, A , P) such that, for every i ∈ {1, 2} and every... integrable random set with values in c(X) satisfying E F log+ F < ∞ and let θ1 , θ2 be two commutative measure- preserving transformations of (Ω, A , P) such that, for every i ∈ {1, 2} and every