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

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

2 Preliminaries

Throughout this paper, let (Ω,A ,P) be a complete probability space, (X, k.k) be a separable Banach space andX∗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 ofX 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 kAk

of A and the support function s(., A) of A are defined by

d (x, A) = inf{kx − yk : y ∈ A},(x ∈X),

kAk = sup{||x|| : x ∈ A},

s(x∗, A) = sup{〈x∗, y〉 : y ∈ A},(x∗∈X∗)

LetP (X) be the family of all nonempty subsets ofX InP (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.

LetBXbe the Borelσ-field onXandBc( )be theσ-field on c(X) generated by the sets

U−= {C ∈ c(X) : C ∩U 6= ;}

taken for all open subsets U ofX

A mapping F from Ω to c(X) is said to be measurable if F is (A ,Bc( ))-measurable, i.e., for every open

set U ofX, the subset

F−1(U−) := {ω ∈ Ω : F (ω) ∩U 6= ;}

belongs toA 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 →Xsuch

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 A F of A by AF = {F−1(U ) : U ∈ Bc( )}, where

F−1(U ) = {ω ∈ Ω : F (ω) ∈ U }, i.e., A F 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 →Xis 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 (equivalence classes of )F -measurable random elements f : Ω →Xsuch that the norm kf k p = (Ek f k p)p1

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),

S p F(F ) = ©f ∈ L p(Ω,F ,P,X) : f ( ω) ∈ F (ω), a.s.ª.

IfF = A then S p

F(F ) is denoted for shortly by S p

F

A random set F : Ω → c(X) is called integrable if the set S1F is nonempty, and it is called integrable bounded

if the random variable kF k is in L1

For any sub-σ-field F of A and any F -measurable random set F , the expectation of F over Ω, with

respect toF , is defined by

E(F, F ) = ©Ef : f ∈ S1

F(F )ª,

where E f =R

Ωf 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 = (n1, n2, , 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{n1, n2, , n d}

Let t be a topology onXand {An : n ∈ N } be an array in c(X) We put

t - lim inf

nmin →∞An=

½

x ∈X: x = t- lim

nmin →∞xn, xn∈ An , ∀n ∈ Nd

¾ ,

t - lim sup

nmin →∞

An=

½

x ∈X: x = t- lim

kmin→∞xk, xk∈ An k , ∀k ∈ Nd

¾

where {An k : k ∈ Nd } is a sub-array of {An : n ∈ Nd}

The sets t - lim infnmin→∞An and t - lim supn

min →∞An are the lower limit and the upper limit of {An : n ∈ Nd }, relative to topology t We obviously have t - lim infnmin→∞An⊂ t - lim supnmin →∞An

An array© An : n ∈ Nd ª converges to A, in the sense of Kuratowski, relatively to the topology t, if the two following equalities are satisfied: t - lim supnmin→∞An= t - lim infnmin→∞An= A In this case, we shall write

A = t-limnmin→∞An; this is true if and only if the next two inclusions hold

t - lim sup

nmin →∞

An⊂ A ⊂ t - lim infn

min →∞An

Let us denote by s (resp w ) the strong (resp. weak) topology of X It is easily seen that

s- lim infnmin→∞An⊂ w- lim supnmin →∞Anand s- lim infnmin→∞An∈ c(X) unless it is empty

A subset A is said to be the Mosco limit of the array {An : n ∈ Nd} denoted by M- limnmin→∞An= A if

w - lim supn

min →∞An= s- lim infnmin →∞An= A which is true if and only if

w - lim sup

nmin→∞

An⊂ A ⊂ s- lim infn

min →∞An

The Wijsman convergence on c(X) is the pointwise convergence of distance functions This means that an array© An : n ∈ Ndª of nonempty closed subsets ofXconverges to A ∈ c(X) with respect to Wijsman convergence, denoted by W- limnmin→∞An= A as nmin→ ∞ if, for every x ∈X, one has

d (x, A) = lim

nmin→∞d (x, An)

The corresponding definitions of pointwise convergence and almost sure convergence for an array

©Fn : n ∈ Ndª of multivalued functions defined onΩ are clear In fact, in the above definitions, it suffices

to replace Anby Fn(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 ofXis Fr´echet differentiable An other interesting feature of the Wijsman topologyTW is that the space (c(X),TW) metrizable and separable, and that the Borelσ-field

ofTWis equal toBc( )(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 inXis 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 everyα ∈ (0,1], L α u = ∩ β<α L β u Let F (X) denote the space of fuzzy sets u :X→ [0, 1] such that

(1) u is normal, i.e., the 1-level set L1u 6= ;,

(2) u is upper semicontinuous, that is, for each α ∈ (0,1], the α-level set L α u is a closed subset ofX

We note that the relation L0u = {x ∈X: u(x) ≥ 0} =Xis automatically satisfied

A linear structure in F (X) is defined by the following operations,

(u + v)(x) = sup

y+z=x

min{u(y), v(z)},

(λu)(x) =

½

u( λ−1x) if λ 6= 0,

I{0}(x) if λ = 0,

where u, v ∈ F (X),λ ∈ R Then for each α ∈ (0,1], L α (u + v) = L α (u) + L α (v) and L α(λu) = λL α (u).

A function u in F (X) is called convex if it satisfies

u( λx + (1 − λ)y) ≥ min{u(x),u(y)} for every x, y ∈Xandλ ∈ [0,1].

It is known that u is convex in the above sense iff, for any α ∈ (0,1], the level set L α u is a convex subset of

X

The closed convex hull co u of u ∈ F (X) is defined as follows:

co u(x) = sup©

α ∈ [0,1] : x ∈ co L α uª ,

so that L α (co u) = co L α u for all α ∈ [0,1].

The concept of fuzzy random set as a generalization for a random set was extensively studied by

M L Puri and D A Ralescu [16] A fuzzy-valued random variable (or fuzzy random set) is a mapping

˜

X : Ω → F (X) such that L α X is a random set for every˜ α ∈ (0,1].

The expected value of any fuzzy random set ˜ X , denoted by E ˜ X , is a fuzzy set such that, for every α ∈ (0,1],

L α¡E ˜X ¢ = E¡L α X˜¢

A mapping v : Ω ×X → [0, 1] is called fuzzy random integrand if it is A ⊗ BX-measurable and

v( ω,·) belongs to F (X) for each ω ∈ Ω The multivalued mapping ω 7→ {x ∈X: v( ω,x) ≥ α} (resp.

ω 7→ cl{x ∈X: v( ω,x) > 0}) will be denoted by L α v (resp L0 +v).

A fuzzy random integrand v :Ω ×X→ [0, 1] is called integrable if the fuzzy random set ω 7→ 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 eachω ∈ 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© fn : n ∈ Nd ª of random elements is called uniformly integrable iff E¡kfnkI (kfnk>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 I θofA We say that

θ is an ergodic if I θ is trivial, i.e., whenever A ∈ I θ 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 I θ-measurable An

extended real-valued measurable function f : Ω → R is called θ-finite if P¡E(f |I θ) = ∞¢ = 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 theσ-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, ,θ k 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.,

E³k f k¡log+k f k¢k−1

´

< ∞, the multiple sequence

A m1, ,m k f := 1

m1 m k

m1−1

X

i1=0

· · ·

m k−1

X

i k=0

f ( θ i1

1 .θ i k

k)

converges a.s to E( f |I θ ) as min{m1, , m k } → ∞, where I θ= ∩i =1Iθ i

Moreover, if θ s is ergodic for some s belonging to {1, 2, , k}, the limit function is constant a.s., i.e.,

E( f |I θ ) = E f a.s.

Proof At first, we will prove this lemma for the real-valued functions case We have

A m1, ,m k f ( ω) = 1

m1 m k

Ã

m1 −1

X

i1 =0

· · ·

m k−1−1

X

i k−1=0

m k−2

X

i k=0

f ( θ i1

1 θ i k

k θ k(ω)) + mX1−1

i1 =0

· · ·

m k−1−1

X

i k−1=0

f ( θ i1

1 θ i k−1 k−1(ω))

!

=m k− 1

m k

Ã

1

m1 m k−1 (m k− 1)

m1−1

X

i1=0

· · ·

m k−1−1

X

i k−1=0

m k−2

X

i k=0

f ( θ i1

1 θ i k

k θ k(ω))

!

+m1

k

à 1

m1 m k−1

m1 −1

X

i1 =0

· · ·

m k−1−1

X

i k−1=0

f ( θ i1

1 θ i k−1 k−1(ω))

!

=m k− 1

m k A m1, ,m k−1 ,m k−1f ( θ k(ω)) + 1

m k A m1, ,m k−1 f ( ω). (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, ,m k f ( ω) → f (ω) as min{m1, , m k } → ∞ for all ω ∈ A1 (3.2) and

A m1, ,m k−1 f ( ω) → f1(ω) as min{m1, , m k } → ∞ for all ω ∈ A2, (3.3)

where both f and f1in L1

Sinceθ kis a measure-preserving transformation, i.e., P(θ−1

k A) = P(A) for all A ∈ A , by choosing A = A1

and by putting A3= θ−1k A1, we have P(A3) = 1 Setting B = ∩3i =1 A i , we check that P(B ) = 1 Thus for every

ω ∈ B, in (3.1), letting min{m1, , m k } → ∞, we obtain f (ω) = f (θ k(ω)) It is equivalent to f = f ◦ θ ka.s, and hencef isIθ k-measurable

Since the transformationsθ1,θ2, ,θ k are commutative, the limit function f isIθ i-measurable for every

i = 1, ,k Therefore, we have that f isIθ-measurable whereIθ= ∩k i =1Iθ i

Let A ofIθ Since I AisIθ i -measurable for every i = 1, ,k, it is θ i-invariant which entails

E((A m1, ,m k f )I A) = E

ÃÃ 1

m1 m k

m1 −1

X

i1 =0

· · ·

m k−1

X

i k=0

f ( θ i1

1 θ i k

k)

!

I A

!

= E

à 1

m1 m k

m1−1

X

i1 =0

· · ·

m k−1

X

i k=0

( f I A)(θ i1

1 θ i k

k)

!

By the similar arguments as in proof of [13, Theorem 2.3, page 9], the multiple sequence A m1, ,m k ( 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 |I θ)

Secondly, in the case of Banach space-valued functions, we proceed as follows: For any x ∈Xand B ∈ A ,

by using the BET for real-valued function I B, we have that

°

A m1, ,m k (x I B ) − E(xI B|Iθ)°=°x ¡ A m1, ,m k I B − E(I B|Iθ)¢°

°= kxk¯A m1, ,m k I B − E(I B|Iθ)¯

tends to 0 a.s as min{m1, , 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, ,m k f − E(f |I θ)°≤°A m1, ,m k ( f − h n)°+°A m1, ,m k h n − E(h n|Iθ)°+°E(hn|Iθ ) − E(f |I θ)° (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{m1, , m k} → ∞

Consequently, for every n,

lim sup

min{m1, ,m k}→∞

°

A m1, ,m k f − E(f |I θ)°≤ lim sup

min{m1, ,m k}→∞

°

A m1, ,m k ( f − h n)°+°E(hn|Iθ ) − E(f |I θ)° (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°= kg k ◦ θ i, one has

°

A m1, ,m k ( f − h n)°

≤ A m1, ,m k k f − h nk

By the BET for real-valued functions case A m1, ,m k k f − h nk tends a.s to a limit function with integral

Ek f − hnk 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,θ2be 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) lim sup

min{m,n}→∞

1

mn

m−1

X

i =0

n−1

X

j =0

f ( θ i

1θ2j ) and lim inf

min{m,n}→∞

1

mn

m−1

X

i =0

n−1

X

j =0

f ( θ i

1θ2j ) are θ2-measurable,

(b) if θ1,θ2are commutative and if letIθ= ∩2i =1Iθ i , then lim inf

min{m,n}→∞

1

mn

m−1

X

i =0

n−1

X

j =0

f ( θ i

1θ2j ) ≥ E(f |I θ ) a.s (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) = n − 1

n A m,n−1 f ( θ2(ω)) +1

By using [10, Theorem 1], we get

lim

m→∞ A m f ( ω) = E(f |I θ1) a.s

Combining this with theθ1-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) = limsup min{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 pdefined by

f p(ω) =

½

f ( ω) if f (ω) ≤ p,

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) ≥ liminf

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

Proposition 3.3 Let F be an integrable random set with values in c(X) satisfying E(kF klog+kF k) < ∞ and let

θ1,θ2be two commutative measure-preserving transformations of ( Ω,A ,P) such that, for every i ∈ {1,2} and

every integer s ≥ 1, θ i s is ergodic Then

co EF ⊂ s- lim inf

min{m,n}→∞

1

mncl

m

X

i =1

n

X

j =1

F ( θ i

1θ2j(ω)) a.s.

Proof Let G mn(ω) = 1

mnclPm

i =1

Pn

j =1 F ( θ i

1θ2j(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 f1, , f k ∈ S F1 such that

°1

k

Pk

i =1 E( fi ) − x°< ε.

At first, we will show that

1

k

k

X

i =1

E( f i ) ∈ s- lim inf

min{m,n}→∞ G mn(ω) a.s. (3.8)

is enough to prove the “lim inf” part co EF ⊂ s-liminf min{m,n}→∞ G mn(ω) a.s of Mosco convergence.

Indeed, let z k= 1kPk

i =1 E( fi) SinceXis separable, there exists a countable dense set D co EF of co EF

For each fixed x ( j ) ∈ D co EF and for everyε k= 1k (k ≥ 1), by (3.8), there exists an element z k ofX, which

depends on x ( j )andε k , such that z k ∈ s- lim inf min{m,n}→∞ G mn(ω) a.s Therefore, for each k ≥ 1, there exists a

negligible set N k ∈ A such that z k ∈ s- lim inf min{m,n}→∞ G mn(ω) for all ω ∈ Ω\N k Letting

N =S∞

k=1 N k , then P(N ) = 0 For each ω ∈ N , it follows from the set s-liminf min{m,n}→∞ G mn(ω) is closed,

z k ∈ s- lim inf min{m,n}→∞ G mn(ω) for all k and z k → x ( j ) as k → ∞, that x ( j ) ∈ s- lim inf min{m,n}→∞ G mn(ω).

This means that x ( j ) ∈ s- lim inf min{m,n}→∞ G mn(ω) a.s., for each fixed j ≥ 1 Noting that D co EFis a countable

set, we obtain D co EF ⊂ s- lim inf min{m,n}→∞ G mn(ω) a.s Since the set s-liminf min{m,n}→∞ G mn(ω) is closed for

eachω, by taking the closure of both sides of the above relation, we have coEF ⊂ s-liminf min{m,n}→∞ G mn(ω)

a.s Therefore, the above statement is proved

Fixed k, as in [3, 4], we define f i j ∈ S1F , i , j = 1, ,k as follows:

f i j=

½

f i +j −1 if i + j ≤ k + 1,

f i +j −1−k if i + j > k + 1.

It is easy to check that

1

k

k

X

i =1

E f i= 1

k2

k

X

i =1

k

X

j =1

Next, we define the double array { f i j : i ≥ 1, j ≥ 1} by

f (s−1)k+i ,(t−1)k+j(ω) = f i j(θ (s−1)k+i

1 θ (t −1)k+j

2 (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 nsatisfying

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

m→∞ s m= ∞ and limn→∞ t n= ∞

Furthermore, it is not difficult to show that for allω ∈ Ω, the following equality holds

1

mn

m

X

i =1

n

X

j =1

f i j(ω) = s m t n

mn

k

X

i =1

k

X

j =1

1

s m t n

s m

X

l =1

t n

X

r =1

f (l −1)k+i ,(r −1)k+j(ω)

+ t n

mn

p m

X

i =1

k

X

j =1

1

t n

t n

X

r =1

f s m k+i ,(r −1)k+j(ω) + s m

mn

k

X

i =1

q n

X

j =1

1

s m

s m

X

l =1

f (l −1)k+i ,t n k+j(ω)

mn

p m

X

i =1

q n

X

j =1

f s m k+i ,t n k+j(ω). (3.12) The proof will be performed as follows

Step 2 Claim 1:

s m t n

mn

k

X

i =1

k

X

j =1

1

s m t n

s m

X

l =1

t n

X

r =1

f (l −1)k+i ,(r −1)k+j(ω) → 1

k2

k

X

i =1

k

X

j =1

E fi j a.s as min{m, n} → ∞. (3.13)

For every integer s ≥ 1, f ∈ S1Fand measure-preserving transformationθ, in view of

E¡°

°f ◦ θ s°

log+°

f ◦ θ s°

¢ = E¡¡kf klog+k f k¢ ◦ θ s ¢ = E¡kf klog+k f k¢ ≤ E¡kF klog+kF k¢ < ∞, 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 transformationsθ k

1,θ k

2and the function f i j ◦ θ1i belonging to the Zygmund’s class, we have

1

s m t n

s m

X

l =1

t n

X

r =1

f (l −1)k+i ,(r −1)k+j(ω) = 1

s m t n

s m

X

l =1

t n

X

r =1

f i j³θ (l −1)k+i

1 θ (r −1)k+j

2

´ (ω)

s m t n

s m

X

l =1

t n

X

r =1

³

f i j ◦ θ1i

´ ³ (θ k

1)l −1(θ k

2)r −1´◦ θ2j(ω)

→ E

³

f i j ◦ θ1i

¯

Iθ k

´

◦ θ2j(ω) a.s as min{m,n} → ∞ where I θ k= ∩2i =1Iθ k

i

= E³f i j ◦ θ i1´◦ θ2j(ω) a.s as min{m,n} → ∞ (by θ k

i is ergodic for some i )

We have

lim

m→∞

s m

m→∞(1

k+ 1

m−p m

mk) = 1

k and limn→∞

t n

n = lim

n→∞(1

k+1

n−q n

nk) =1

Therefore, (3.13) holds

Claim 2:

t n

mn

p m

X

i =1

k

X

j =1

1

t n

t n

X

r =1

f s m k+i ,(r −1)k+j(ω) + s m

mn

k

X

i =1

q n

X

j =1

1

s m

s m

X

l =1

f (l −1)k+i ,t n k+j(ω) → 0 a.s as min{m,n} → ∞ (3.16)

For each m ≥ 1 and i , j ∈ {1, ,k}, by applying Lemma 3.1 for the measure-preserving transformation θ k2 and the function f i j ◦ θ s m k+i

1 in L1(X), we get 1

t n

t n

X

r =1

f s m k+i ,(r −1)k+j(ω) = 1

t n

t n

X

r =1

f i j³θ s m k+i

1 θ (r −1)k+j

2

´ (ω)

=t1

n

t n

X

r =1

³

f i j ◦ θ s m k+i

1

´ (θ k

2)r −1 ◦ θ2j(ω)

→ E³f i j ◦ θ s m k+i

1

¯

Iθ k

´

◦ θ2j(ω) a.s as n → ∞ (by Lemma 3.1)

= E

³

f i j ◦ θ s m k+i

1

´

◦ θ2j(ω) a.s as n → ∞ (by θ k

2is ergodic)

Thus, we have that

t n mn

p m

X

i =1

k

X

j =1

1

t n

t n

X

r =1

f s m k+i ,(r −1)k+j(ω) → 0 a.s as min{m,n} → ∞.

Similarly, we obtain

s m mn

k

X

i =1

q n

X

j =1

1

s m

s m

X

l =1

f (l −1)k+i ,t n k+j(ω) → 0 a.s as min{m,n} → ∞.

Hence, (3.16) is proved

Claim 3:

1

mn

p m

X

i =1

q n

X

j =1

f s m k+i ,t n k+j(ω) → 0 a.s as min{m,n} → ∞. (3.18)

From (3.14), (3.15) and (3.17), we deduce that for each i , j ∈ {1, ,k},

1

mn f s m k+i ,t n k+j(ω) = s m t n

mn

à 1

s m t n

s m

X

l =1

t n

X

r =1

f l k+i ,r k+j(ω) − s m− 1

s m .

1

(s m − 1)t n

s m−1

X

l =1

t n

X

r =1

f l k+i ,r k+j(ω)

−t s n− 1

m t n.

1

t n− 1

t n−1

X

r =1

f s m k+i ,r k+j(ω)

!

k2(E f i j − 1.E f i j − 0.E f i j ) = 0 a.s as min{m,n} → ∞,

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

mn

m

X

i =1

n

X

j =1

f i j(ω) → 1

k

k

X

i =1

E fi a.s as min{m, n} → ∞.

Note that from the inclusion

1

mn

m

X

i =1

n

X

j =1

f i j(ω) ∈ 1

mn

m

X

i =1

n

X

j =1

F ( θ i

1θ2j(ω)) a.s.,

it follows from (3.8) that

1

k

k

X

i =1

E f i ∈ s- lim inf

min{m,n}→∞

1

mncl

m

X

i =1

n

X

j =1

F ( θ i

1θ2j(ω)) a.s.

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, ,θ k 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 E³kF k¡log+kF k¢k−1´

< ∞, we have

E(F |Iθ ) ⊂ s- lim inf

min{m1, ,m k}→∞

1

m1 m kcl

m1−1

X

i1=0

· · ·

m k−1

X

i k=0

F ( θ i1

1 θ i k

k(ω)) a.s., whereIθ= ∩k i =1Iθ i

Proof Let A m1, ,m k F ( ω) = 1

m1 m kclPm1 −1

i1 =0 · · ·P k−1

i k=0 F ( θ i1

1 θ k

k(ω)), ω ∈ Ω By virtue of [18, Lemma 2.2],

we can choose { f n : n ≥ 1} as a sequence in S1F such that©E(f n|Iθ ) : n ≥ 1ª be a Castaing representation of

E(F |Iθ) whereIθ= ∩k i =1Iθ i According to Lemma 3.1 for the functions f n , n ≥ 1 belonging to Zygmund’s

class, we get

lim

min{m1, ,m k}→∞A m1, ,m k 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( fn|Iθ)(ω) ∈ s- lim inf

min{m1, ,m k}→∞A m1, ,m k F ( ω) for all integer n ≥ 1.

Since the E( f n|Iθ)(ω), n ≥ 1 are dense in E(F |I θ)(ω) and s-liminf min{m1, ,m k}→∞A m1, ,m k F ( ω) is closed,

we obtain

E(F |I θ ) ⊂ s- lim inf

min{m1, ,m k}→∞A m1, ,m k 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(kF klog+kF k) < ∞ and let

θ1,θ2be two commutative measure-preserving transformations of ( Ω,A ,P) such that θ i is ergodic for some

i ∈ {1,2} Then

w - lim sup

min{m,n}→∞

1

mncl

m

X

i =1

n

X

j =1

F ( θ i

1θ j

2(ω)) ⊂ co EF a.s.

Proof Put G mn(ω) = 1

mnclPm

i =1

Pn

j =1 F ( θ i

1θ j

2(ω)), m,n ≥ 1, ω ∈ Ω Let {x k : k ≥ 1} be a dense sequence of

X\co EF SinceXis separable, by the separation theorem, there exists a sequence©x∗

k : k ≥ 1ª inX∗with

°

x∗

k

°

= 1 such that

〈x k∗, x k 〉 − d ¡x k , co EF ¢ ≥ s ¡x∗

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¡kF klog+kF k¢ < ∞ for every k ≥ 1. (3.20)

So, according to Lemma 3.1 for real-valued case, there exists a negligible subset N ofA such that for every

ω ∈ Ω\N and k ≥ 1,

s ¡x∗

k ,G mn(ω)¢ = 1

mn

m

X

i =1

n

X

j =1

s³x∗k , F

³

θ i

1θ2j(ω)´´→ s ¡x∗

k , co EF ¢ < ∞ as min{m,n} → ∞.

If y ∈ w-limsup min{m,n}→∞ G mn(ω) for ω ∈ Ω\N, then there exists y r s ∈ G m r n s(ω) such that

〈x∗k , y〉 = lim

min{r,s}→∞ 〈x k∗, y r s〉 ≤ lim

min{r,s}→∞ s ¡x∗

k ,G m r n s(ω)¢ = s ¡x∗

k , co EF¢

for all k ≥ 1,

which implies y ∈ co EF Thus w-limsup min{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