Convergence of Weak*-Scalarly Integrable Functions

Abstract

Let $(\Omega ,\mathcal{F},\mu )$ be a complete probability space, E a separable Banach space and $E^{\prime }$ the topological dual vector space of E. We present some compactness results in $L_{E^{\prime }}^1\left[E\right]$, the Banach space of weak*-scalarly integrable $E^{\prime }$-valued functions. As well we extend the classical theorem of Komlós to the bounded sequences in $L_{E^{\prime }}^1\left[E\right]$.

1. Introduction

In their 2001 paper, Benabdellah and Castaing [1] established that the Ülger–Diestel–Ruess–Shachermayer characterization for weak compactness in $L_E^1\left(\mu \right)$ can be extended to $L_{E^{\prime }}^1\left[E\right]$. In addition, they gave several results on weak compactness and conditionally weak compactness in $L_{E^{\prime }}^1\left[E\right]$. These results are not standard and rely on a Talagrand decomposition type theorems for bounded sequences in $L_{E^{\prime }}^1\left[E\right]$. Moreover, this paper paved the way for many researchers to exploit and establish further interesting results in this space (see [2,3,4]).

In this paper, we aim to present some compactness results in $L_{E^{\prime }}^1\left[E\right]$ and a Komlós theorem in $L_{E^{\prime }}^1\left[E\right]$. More precisely, we will give in the first part a decomposition theorem for a bounded sequence in $L_{E^{\prime }}^1\left[E\right]$ (Theorem 2 (i)) and a Komlós-type result for the weak* convergence in $E^{\prime }$ (Theorem 2 (ii)). This will allow us to state a criterion for the $\sigma (L_{E^{\prime }}^1\left[E\right],L_E^{\infty }\left(\mu \right))$ compactness in $L_{E^{\prime }}^1\left[E\right]$ (Theorem 3). In the second part we give a result on weak compactness in $L_{E^{\prime }}^1\left[E\right]$ (Theorem 4 (jj)) in terms of a Komlós theorem for the weak convergence in $E^{\prime }$ (Theorem 4 (j)). Corollary 1 provides a compactness criterion in $L_{E^{\prime }}^1\left[E\right]$, which generalizes Proposition 5.1 in [1]. In this paper, we have established a Komlós theorem in $L_{E^{\prime }}^1\left[E\right]$ and used it to give some weak convergence results. Other works have followed a similar approch in different function spaces, such as the space of Bochner integrable functions and the space of Pettis integrable functions (see [5,6,7]).

2. Notations and Preliminaries

Throughout this paper the triple $(\Omega ,\mathcal{F},\mu )$ is a complete probability space, E is a separable Banach space and $E^{\prime }$ is its topological dual. The weak topology $\sigma (E^{\prime },E^{″})$ (resp. the weak* topology $\sigma (E^{\prime },E)$) on $E^{\prime }$ will be referred to by the symbol “w” (resp. w*). A mapping $f:\Omega \to E^{\prime }$ is w*-measurable, if for any $x\in E$, the function $\langle f,x\rangle :\omega \mapsto \langle f\left(\omega \right),x\rangle$ is $\mathcal{F}$-measurable. Two w*-measurable mappings f and g are said to be equivalent (shortly $f\equiv g\left(w^{*}\right))$ iff $\langle f,x\rangle =\langle g,x\rangle \mu$-$a.e$$\forall x\in E$. Let $L_E^1\left(\mu \right)$ denotes the set of all (equivalence classes of) Bochner integrable E-valued functions [8], recall that (see [9]) the dual of $L_E^1\left(\mu \right)$ is the (quotient) space $L_{E^{\prime }}^{\infty }\left[E\right]$ of w*-measurable bounded functions from $\Omega$ into $E^{\prime }$. Now, according to [4], the set $L_{E^{\prime }}^1(\Omega ,\mathcal{F},\mu ,\left[E\right])$, in short $L_{E^{\prime }}^1\left[E\right]$, denotes the (quotient) space of all w*-measurable mappings $f:\Omega \to E^{\prime }$, such that $\omega \mapsto \langle f\left(\omega \right),x\rangle$ is integrable $\forall x\in E$ and ${\parallel f(.)\parallel }_{E^{\prime }}$ belongs to $L_{\mathbb{R}}^1\left(\mu \right)$, and the mapping

$${\overline{N}}_1\left(f\right)={\int }_{\Omega }\parallel f\parallel d\mu ,f\in L_{E^{\prime }}^1\left[E\right]$$

defines a norm in $L_{E^{\prime }}^1\left[E\right]$. Furthermore, the set $L_E^{\infty }\left(\mu \right)$ of all (equivalence classes of) $\mu$-measurable essentially bounded functions with value in E is included in the topological dual of $L_{E^{\prime }}^1\left[E\right]$ and the mapping $f\mapsto {\overline{N}}_1\left(f\right)$ is lower semicontinuous on $L_{E^{\prime }}^1\left[E\right]$ for the topology $\sigma (L_{E^{\prime }}^1\left[E\right],L_E^{\infty }\left(\mu \right))$.

In addition, recall that $(L_{E^{\prime }}^1\left[E\right],{\overline{N}}_1)$ is a Banach space ([1], Proposition 3.4) and that a subset $\mathcal{K}$ of $L_{E^{\prime }}^1\left[E\right]$ is uniformly integrable (briefly UI) if the set $\{\parallel f\parallel ;f\in \mathcal{K}\}$ is UI in $L_{\mathbb{R}}^1\left(\mu \right)$ ([1], Definition 4.2). A subset $\mathcal{K}$ of $L_{\mathbb{R}}^1\left(\mu \right)$ is UI if

$$\underset{t\to \infty }{lim}\underset{f\in \mathcal{K}}{sup}{\int }_{\left\{\right|f|\ge t\}}\left|f\right|d\mu =0.$$

Note that every UI subset of $L_{E^{\prime }}^1\left[E\right]$ is ${\overline{N}}_1$-bounded.

Finally let us recall the notion of the K-convergence [5]. Let ${\left(f_n\right)}_{n\in \mathbb{N}}$ a sequence from $\Omega$ to $E^{\prime }$ and F be a subset of $E^{″}$. We say that ${\left(f_n\right)}_{n\in \mathbb{N}}$ is $\sigma (E^{\prime },F)$-K converge almost everywhere on $\Omega$ to a function f if for every subsequence ${\left(f_n^{\prime }\right)}_{n\in \mathbb{N}}$ of ${\left(f_n\right)}_{n\in \mathbb{N}}$ there exists a null set $\mathcal{N}\in \mathcal{F}$, such that for every $\omega \in \Omega \backslash \mathcal{N}$

$$\forall x\in F,\langle x,\frac{1}{n}{\displaystyle \sum _{i=1}^n}{f}_i^{\prime }\left(\omega \right)\rangle \to \langle x,f\left(\omega \right)\rangle .$$

A well-known theorem of Komlós is as follows:

Theorem 

([10]). Every bounded sequence in $L_{\mathbb{R}}^1\left(\mu \right)$ has a subsequence which K-converges $a.e.$ to a real integrable function.

For some K-convergence results in infinite dimension we can see [11,12,13,14,15], and for more details and results on $L_{E^{\prime }}^1\left[E\right]$, we refer to [1,2,3,4,9].

3. Main Results

We begin by recalling the following result ([16], Lemma 4.1) which is important for the development of the work.

Lemma 1.

Let ${\left(f_n\right)}_{n\in \mathbb{N}}$ be a bounded sequence in $L_E^1\left(\mu \right)$. Then, there exists a subsequence ${\left(g_n\right)}_{n\in \mathbb{N}}$ of ${\left(f_n\right)}_{n\in \mathbb{N}}$, such that for every subsequence ${\left(h_n\right)}_{n\in \mathbb{N}}$ of ${\left(g_n\right)}_{n\in \mathbb{N}}$

(a) 

The sequence ${\left(1_{\left\{\parallel h_n\parallel <n\right\}}{h}_n\right)}_{n\in \mathbb{N}}$ is uniformly integrable;

(b) 

The sequence ${(h_n-1_{\left\{\parallel h_n\parallel <n\right\}}{h}_n)}_{n\in \mathbb{N}}$ converges $a.e$ to 0 in $E.$

Let ${\left(f_n\right)}_{n\in \mathbb{N}}$ be a bounded sequence in $L_{E^{\prime }}^1\left[E\right]$, as the sequence $(\parallel f_n{\parallel )}_{n\in \mathbb{N}}$ is bounded in $L_{\mathbb{R}}^1\left(\mu \right)$, by Lemma 1 there exists a subsequence $(\parallel g_n{\parallel )}_{n\in \mathbb{N}}$ of $(\parallel f_n{\parallel )}_{n\in \mathbb{N}}$, such that $(1_{\left\{\parallel h_n\parallel <n\right\}}\parallel h_n{\parallel )}_{n\in \mathbb{N}}$ is UI and $(\parallel h_n\parallel -1_{\left\{\parallel h_n\parallel <n\right\}}\parallel h_n{\parallel )}_{n\in \mathbb{N}}$ converges $a.e$ to 0 in $\mathbb{R}$ for each subsequence $(\parallel h_n{\parallel )}_{n\in \mathbb{N}}$. Then, we can see that the sequences ${\left(g_n\right)}_{n\in \mathbb{N}}$ and ${\left(h_n\right)}_{n\in \mathbb{N}}$ have the required properties of the next lemma.

Lemma 2.

Let ${\left(f_n\right)}_{n\in \mathbb{N}}$ be a bounded sequence in $L_{E^{\prime }}^1\left[E\right]$. Then there exists a subsequence ${\left(g_n\right)}_{n\in \mathbb{N}}$ of ${\left(f_n\right)}_{n\in \mathbb{N}}$, such that for every subsequence ${\left(h_n\right)}_{n\in \mathbb{N}}$ of ${\left(g_n\right)}_{n\in \mathbb{N}}$

(a′) 

The sequence ${\left(1_{\left\{\parallel h_n\parallel <n\right\}}{h}_n\right)}_{n\in \mathbb{N}}$ is uniformly integrable;

(b′) 

The sequence ${(h_n-1_{\left\{\parallel h_n\parallel <n\right\}}{h}_n)}_{n\in \mathbb{N}}$ converges $a.e$ to 0 in $E^{\prime }.$

The following simple result is useful.

Lemma 3.

Every bounded set in $L_{E^{\prime }}^{\infty }\left[E\right]$ is sequentially relatively compact for the topology $\sigma (L_{E^{\prime }}^{\infty }\left[E\right],L_E^{\infty }\left(\mu \right))$.

Proof. 

Let H be a bounded set in $L_{E^{\prime }}^{\infty }\left[E\right]={\left(L_E^1\left(\mu \right)\right)}^{\prime }$, by the Banach–Alaoglu theorem H is relatively compact for the topology $\sigma (L_{E^{\prime }}^{\infty }\left[E\right],L_E^1\left(\mu \right))$. As $L_E^1\left(\mu \right)$ is separable because E it is, H is sequentially relatively compact for the topology $\sigma (L_{E^{\prime }}^{\infty }\left[E\right],L_E^1\left(\mu \right))$, and since $L_E^{\infty }\left(\mu \right)$ is a subspace of $L_E^1\left(\mu \right)$, we deduce that H is $\sigma (L_{E^{\prime }}^{\infty }\left[E\right],L_E^{\infty }\left(\mu \right))$-sequentially relatively compact. □

Lemma 4.

Let ${\left(f_n\right)}_{n\in \mathbb{N}}$ be a sequence of $L_{E^{\prime }}^1\left[E\right]$ which converges $\sigma (L_{E^{\prime }}^1\left[E\right],L_E^{\infty }\left(\mu \right))$ to a function $f\in L_{E^{\prime }}^1\left[E\right]$. Then, there exists an integer m such that

$${\overline{N}}_1\left(f\right)\le 2\underset{n\ge m}{inf}{\overline{N}}_1\left(f_n\right).$$

Proof. 

As the mapping ${\overline{N}}_1$ is lower semicontinuous on $L_{E^{\prime }}^1\left[E\right]$ for the topology $\sigma (L_{E^{\prime }}^1\left[E\right],L_E^{\infty }\left(\mu \right))$, we have ${\overline{N}}_1\left(f\right)\le {\displaystyle \underset{n}{lim}}inf{\overline{N}}_1\left(f_n\right)$. If ${\displaystyle \underset{n}{lim}}inf{\overline{N}}_1\left(f_n\right)=0$, then the result is obvious. Now, if ${\displaystyle \underset{n}{lim}}inf{\overline{N}}_1\left(f_n\right)>0$, we have

$${\overline{N}}_1\left(f\right)<2{\displaystyle \underset{n}{lim}}inf{\overline{N}}_1\left(f_n\right)=\underset{m\ge 1}{sup}2\underset{n\ge m}{inf}{\overline{N}}_1\left(f_n\right).$$

Hence there exists $m\in {\mathbb{N}}^{*}$, satisfying the inequality. □

Now we are able to state our first main result of this paper.

Theorem 2.

Let ${\left(f_n\right)}_{n\in \mathbb{N}}$ be a bounded sequence in $L_{E^{\prime }}^1\left[E\right]$. Then, there exists a function f in $L_{E^{\prime }}^1\left[E\right]$ and a subsequence ${\left(g_n\right)}_{n\in \mathbb{N}}$ of ${\left(f_n\right)}_{n\in \mathbb{N}}$ such that for every subsequence ${\left(h_n\right)}_{n\in \mathbb{N}}$ of ${\left(g_n\right)}_{n\in \mathbb{N}}$ the following holds

(i) 

${\left(1_{\left\{\parallel h_n\parallel <n\right\}}h\right)}_{n\in \mathbb{N}}$ converges $\sigma (L_{E^{\prime }}^1\left[E\right],L_E^{\infty }\left(\mu \right))$ to f in $L_{E^{\prime }}^1\left[E\right]$ and ${(h_n-1_{\left\{\parallel h_n\parallel <n\right\}}h)}_{n\in \mathbb{N}}$ converges $a.e.$ to 0 in $E^{\prime }$;

(ii) 

${(\frac{1}{n}{\displaystyle \sum _{i=1}^n}{h}_i)}_{n\in \mathbb{N}}\mathit{w}*-\mathit{converges}a.e.\mathit{to}f.$

Proof. 

(i) We have $\parallel 1_{\left\{\parallel f_n\parallel <k\right\}}{f}_n{\parallel }_{L_{E^{\prime }}^{\infty }\left[E\right]}\le k$ for all $(k,n)\in {\left({\mathbb{N}}^{*}\right)}^2$. For $k=1$, there exists by Lemma 3 a subsequence ${\left(f_n^1\right)}_{n\in \mathbb{N}}$ of ${\left(f_n\right)}_{n\in \mathbb{N}}$, such that the sequence ${\left(1_{\left\{\parallel f_n^1\parallel <1\right\}}{f}_n^1\right)}_{n\in \mathbb{N}}$ converges $\sigma (L_{E^{\prime }}^{\infty }\left[E\right],L_E^{\infty }\left(\mu \right))$ to $v_1\in L_{E^{\prime }}^{\infty }\left[E\right]$ and there exists for all $k\ge 1$ a subsequence ${\left(f_n^{k+1}\right)}_{n\in \mathbb{N}}$ of ${\left(f_n^k\right)}_{n\in \mathbb{N}}$, such that the sequence ${\left(1_{\left\{\parallel f_n^{k+1}\parallel <k+1\right\}}{f}_n^{k+1}\right)}_{n\in \mathbb{N}}$ converges $\sigma (L_{E^{\prime }}^{\infty }\left[E\right],L_E^{\infty }\left(\mu \right))$ to $v_{k+1}$ in $L_{E^{\prime }}^{\infty }\left[E\right]$. Let $f_n^{\prime }=f_n^n(n\ge 1)$, then, for every $k\ge 1$, the sequence ${\left(1_{\left\{\parallel f_n^{\prime }\parallel <k\right\}}{f}_n^{\prime }\right)}_{n\in \mathbb{N}}$ converges $\sigma (L_{E^{\prime }}^{\infty }\left[E\right],L_E^{\infty }\left(\mu \right))$ to $v_k$ in $L_{E^{\prime }}^{\infty }\left[E\right]$.

Claim:${\left(v_k\right)}_{k\in \mathbb{N}}$ converges to a function f in $L_{E^{\prime }}^1\left[E\right]$.

Put $v_0=0$, as $(L_{E^{\prime }}^1\left[E\right],{\overline{N}}_1)$ is a Banach space, it is enough to prove that the series ${\displaystyle \sum _{k\ge 1}}{\overline{N}}_1(v_k-v_{k-1})$ converges. For every $k\ge 1$, the sequence ${(1_{\left\{\parallel f_n^{\prime }\parallel <k\right\}}{f}_n^{\prime }-1_{\left\{\parallel f_n^{\prime }\parallel <k-1\right\}}{f}_n^{\prime })}_{n\in \mathbb{N}}$ converges $\sigma (L_{E^{\prime }}^{\infty }\left[E\right],L_E^{\infty }\left(\mu \right))$ to $(v_k-v_{k-1})$ in $L_{E^{\prime }}^{\infty }\left[E\right]$ and, therefore, also in $L_{E^{\prime }}^1\left[E\right]$ for the topology $\sigma (L_{E^{\prime }}^1\left[E\right],L_E^{\infty }\left(\mu \right))$. By Lemma 4, there exists $m_k\in {\mathbb{N}}^{*}$, such that

$${\overline{N}}_1(v_k-v_{k-1})\le 2\underset{n\ge m_k}{inf}{\overline{N}}_1(1_{\left\{\parallel f_n^{\prime }\parallel <k\right\}}{f}_n^{\prime }-1_{\left\{\parallel f_n^{\prime }\parallel <k-1\right\}}{f}_n^{\prime }).$$

Let $N\in {\mathbb{N}}^{*}$ and $n\ge max(m_1,..,m_N)$. Then we have

$$\begin{array}{cc}\hfill \stackrel{N}{\sum _{k=1}}{\overline{N}}_1(v_k-v_{k-1})& \le 2\stackrel{N}{\sum _{k=1}}{\overline{N}}_1(1_{\left\{\parallel f_n^{\prime }\parallel <k\right\}}{f}_n^{\prime }-1_{\left\{\parallel f_n^{\prime }\parallel <k-1\right\}}{f}_n^{\prime })\hfill \\ \hfill & =2\stackrel{N}{\sum _{k=1}}\int \parallel 1_{\left\{\parallel f_n^{\prime }\parallel <k\right\}}{f}_n^{\prime }-1_{\left\{\parallel f_n^{\prime }\parallel <k-1\right\}}{f}_n^{\prime })\parallel d\mu \hfill \\ \hfill & \le 2\int \parallel f_n^{\prime }\parallel d\mu \hfill \\ \hfill & \le 2\underset{p\ge 1}{sup}{\overline{N}}_1\left(f_p^{\prime }\right)<+\infty \hfill \end{array}$$

and therefore ${\displaystyle \sum _{k=1}^{+\infty }}{\overline{N}}_1(v_k-v_{k-1})<+\infty .$ This proves the Claim.

Now applying Lemma 2 to ${\left(f_n^{\prime }\right)}_{n\in \mathbb{N}}$ we get a subsequence ${\left(f_n^{″}\right)}_{n\in \mathbb{N}}$ of ${\left(f_n^{\prime }\right)}_{n\in \mathbb{N}}$ such that for every subsequence ${\left(h_n\right)}_{n\in \mathbb{N}}$ of ${\left(f_n^{″}\right)}_{n\in \mathbb{N}}$

$${\left(1_{\left\{\parallel h_n\parallel <n\right\}}{h}_n\right)}_{n\in \mathbb{N}}\mathrm{is}\mathrm{UI},$$

(1)

$${(h_n-1_{\left\{\parallel h_n\parallel <n\right\}}{h}_n)}_{n\in \mathbb{N}}\mathrm{converges}a.e.\mathrm{to}0\mathrm{in}{E}^{\prime }.$$

(2)

It remains to show that ${\left(1_{\left\{\parallel h_n\parallel <n\right\}}{h}_n\right)}_{n\in \mathbb{N}}$ converges $\sigma (L_{E^{\prime }}^1\left[E\right],L_E^{\infty }\left(\mu \right))$ to f in $L_{E^{\prime }}^1\left[E\right]$. Let us consider $\zeta \in L_E^{\infty }\left(\mu \right)$ with norm $\le 1$ and $ϵ>0$. By (1) and the convergence of ${\left(v_k\right)}_{k\in \mathbb{N}}$ to f in $L_{E^{\prime }}^1\left[E\right]$, there exists $n_0\in \mathbb{N}$, such that

$$\underset{n}{sup}{\int }_{\left\{\parallel 1_{\left\{\parallel h_n\parallel <n\right\}}{h}_n\parallel \ge n_0\right\}}\parallel 1_{\left\{\parallel h_n\parallel <n\right\}}{h}_n\parallel d\mu =\underset{n}{sup}{\overline{N}}_1(1_{\left\{\parallel h_n\parallel <n\right\}}{h}_n-1_{\left\{\parallel h_n\parallel <n_0\right\}}{h}_n)\le \frac{ϵ}{3}$$

and

$${\overline{N}}_1(v_{n_0}-f)\le \frac{ϵ}{3}.$$

As ${\left(1_{\left\{\parallel h_n\parallel <n_0\right\}}{h}_n\right)}_{n\in \mathbb{N}}$ converges $\sigma (L_{E^{\prime }}^1\left[E\right],L_E^{\infty }\left(\mu \right))$ to $v_{n_0}$ in $L_{E^{\prime }}^1\left[E\right]$, there exists $n_1\ge n_0$ such that

$$n\ge n_1\Rightarrow 〈\zeta ,1_{\left\{\parallel h_n\parallel <n_0\right\}}{h}_n-v_{n_0}〉\le \frac{ϵ}{3}.$$

Then, for $n\ge n_1$ we have

$$\begin{array}{cc}\hfill 〈\zeta ,1_{\left\{\parallel h_n\parallel <n\right\}}{h}_n-f〉& \le 〈\zeta ,1_{\left\{\parallel h_n\parallel <n\right\}}{h}_n-1_{\left\{\parallel h_n\parallel <n_0\right\}}{h}_n〉\hfill \\ \hfill & +〈\zeta ,1_{\left\{\parallel h_n\parallel <n_0\right\}}{h}_n-v_{n_0}〉+〈\zeta ,v_{n_0}-f〉\hfill \\ \hfill & \le {\overline{N}}_1(1_{\left\{\parallel h_n\parallel <n\right\}}{h}_n-1_{\left\{\parallel h_n\parallel <n_0\right\}}{h}_n)\hfill \\ \hfill & +〈\zeta ,1_{\left\{\parallel h_n\parallel <n_0\right\}}{h}_n-v_{n_0}〉+{\overline{N}}_1(v_{n_0}-f)\hfill \\ \hfill & \le ϵ.\hfill \end{array}$$

(ii) It is sufficient to show that there is a subsequence ${\left(g_n\right)}_{n\in \mathbb{N}}$ of ${\left(f_n^{″}\right)}_{n\in \mathbb{N}}$, such that ${(\frac{1}{n}{\displaystyle \sum _{i=1}^n}{h}_i)}_{n\in \mathbb{N}}$ w*-converges $a.e.$ to f for every subsequence ${\left(h_n\right)}_{n\in \mathbb{N}}$ of ${\left(g_n\right)}_{n\in \mathbb{N}}$. With E being separable, let $D={\left(x_j\right)}_{j\in {\mathbb{N}}^{*}}$, a norm-dense sequence in E. The sequences $(\parallel f_n^{″}(.){\parallel )}_{n\in \mathbb{N}}$ and ${\left(\langle f_n^{\prime \prime },x_j\rangle \right)}_{n\in \mathbb{N}}$$j=1,2,....$ are bounded in $L_{\mathbb{R}}^1\left(\mu \right)$, so we apply Komlós’ theorem to suitably chosen sequences and a diagonal method to get functions ${\phi }_0$,${\phi }_1$,${\phi }_2$, …, ${\phi }_j,...$ in $L_{\mathbb{R}}^1\left(\mu \right)$ and a subsequence ${\left(g_n\right)}_{n\in \mathbb{N}}$ of ${\left(f_n^{″}\right)}_{n\in \mathbb{N}}$, such that for every subsequence ${\left(h_n\right)}_{n\in \mathbb{N}}$ of ${\left(g_n\right)}_{n\in \mathbb{N}}$

$$\frac{1}{n}{\displaystyle \sum _{i=1}^n}\parallel h_i\left(\omega \right)\parallel \to {\phi }_0\left(\omega \right)a.e.,$$

(3)

$$\forall j\in {\mathbb{N}}^{*},\frac{1}{n}{\displaystyle \sum _{i=1}^n}\langle h_i\left(\omega \right),x_j\rangle \to {\phi }_j\left(\omega \right)a.e.$$

(4)

Let ${\left(h_n\right)}_{n\in \mathbb{N}}$ be a fixed subsequence of ${\left(g_n\right)}_{n\in \mathbb{N}}$. By (2) and the decomposition $1_{\left\{\parallel h_n\parallel <n\right\}}{h}_n=h_n-(h_n-1_{\left\{\parallel h_n\parallel <n\right\}}{h}_n)$ we get

$$\forall j\in {\mathbb{N}}^{*},\frac{1}{n}{\displaystyle \sum _{i=1}^n}\langle 1_{\left\{\parallel h_i\parallel <i\right\}}{h}_i\left(\omega \right),x_j\rangle \to {\phi }_j\left(\omega \right)a.e.$$

(5)

As ${\left(\langle 1_{\left\{\parallel h_n\parallel <n\right\}}{h}_n,x_j\rangle \right)}_{n\in \mathbb{N}}$ is UI for each $j\in {\mathbb{N}}^{*}$, it follows by (5) and the Lebesgue–Vitali’s theorem that for each $A\in \mathcal{F}$

$$\forall j\in {\mathbb{N}}^{*},\frac{1}{n}\stackrel{n}{\sum _{i=1}}{\int }_A\langle 1_{\left\{\parallel h_i\parallel <i\right\}}{h}_i,x_j\rangle d\mu \to {\int }_A{\phi }_{j}d\mu .$$

(6)

On the other hand, by (i)

$$\forall l\in L_E^{\infty }\left(\mu \right),\frac{1}{n}{\displaystyle \sum _{i=1}^n}{\int }_A\langle 1_{\left\{\parallel h_i\parallel <i\right\}}{h}_i,l\rangle d\mu \to {\int }_{\Omega }\langle f,l\rangle d\mu ,$$

(7)

so in particular for each $A\in \mathcal{F}$ and $x_j\in D$ we have

$$\frac{1}{n}{\displaystyle \sum _{i=1}^n}{\int }_A\langle 1_{\left\{\parallel h_i\parallel <i\right\}}{h}_i,x_j\rangle d\mu \to {\int }_A\langle f,x_j\rangle d\mu ,$$

(8)

then by (6) and (8) we get

$$\forall j\in {\mathbb{N}}^{*},{\phi }_j\left(\omega \right)=\langle f\left(\omega \right),x_j\rangle a.e.$$

(9)

and therefore by (4)

$$\forall j\in {\mathbb{N}}^{*},\langle \frac{1}{n}{\displaystyle \sum _{i=1}^n}{h}_i\left(\omega \right),x_j\rangle \to \langle f\left(\omega \right),x_j\rangle a.e.$$

(10)

Finally, by (3), ${(\frac{1}{n}{\displaystyle \sum _{i=1}^n}{h}_i(.))}_{n\in \mathbb{N}}$ is pointwise bounded $a.e.;$ this, along with the density of D, yields

$$\forall x\in E,\langle \frac{1}{n}{\displaystyle \sum _{i=1}^n}{h}_i\left(\omega \right),x\rangle \to \langle f\left(\omega \right),x\rangle a.e.$$

So the proof is complete. □

An immediate application of Theorem 1, we have the following criteria for $\sigma (L_{E^{\prime }}^1\left[E\right],L_E^{\infty }\left(\mu \right))$ compactness in $L_{E^{\prime }}^1\left[E\right]$, which generalizes Lemma 3.

Theorem 3.

Every uniformly integrable set in $L_{E^{\prime }}^1\left[E\right]$ is sequentially relatively compact for the topology $\sigma (L_{E^{\prime }}^1\left[E\right],L_E^{\infty }\left(\mu \right))$.

Proof. 

Let H be an UI set in $L_{E^{\prime }}^1\left[E\right]$ and ${\left(f_n\right)}_{n\in \mathbb{N}}$ a sequence in H. As ${\left(f_n\right)}_{n\in \mathbb{N}}$ is bounded, by Theorem 1 (i) there is a function f in $L_{E^{\prime }}^1\left[E\right]$ and a subsequence ${\left(h_n\right)}_{n\in \mathbb{N}}$ of ${\left(f_n\right)}_{n\in \mathbb{N}}$, such that ${\left(1_{\left\{\parallel h_n\parallel <n\right\}}{h}_n\right)}_{n\in \mathbb{N}}$ converges $\sigma (L_{E^{\prime }}^1\left[E\right],L_E^{\infty }\left(\mu \right))$ to f in $L_{E^{\prime }}^1\left[E\right]$ and ${\left(1_{\left\{\parallel h_n\parallel \ge n\right\}}{h}_n\right)}_{n\in \mathbb{N}}$ converges $a.e.$ to 0 in $E^{\prime }$. As ${\left(h_n\right)}_{n\in \mathbb{N}}$ is UI, ${\left(1_{\left\{\parallel h_n\parallel \ge n\right\}}{h}_n\right)}_{n\in \mathbb{N}}$ converges strongly to 0 in $L_{E^{\prime }}^1\left[E\right]$ and hence ${\left(h_n\right)}_{n\in \mathbb{N}}$ converges $\sigma (L_{E^{\prime }}^1\left[E\right],L_E^{\infty }\left(\mu \right))$ to f in $L_{E^{\prime }}^1\left[E\right]$. Then H is $\sigma (L_{E^{\prime }}^1\left[E\right],L_E^{\infty }\left(\mu \right))$-sequentially relatively compact. □

It is well known that the Komlós type results can be used to develop weak compactness criteria in $L_E^1\left(\mu \right)$. Using this argument, we now provide some weak compactness results in $L_{E^{\prime }}^1\left[E\right]$.

Lemma 5.

Let ${\left(f_n\right)}_{n\in \mathbb{N}}$ be a uniformly integrable sequence in $L_{E^{\prime }}^1\left[E\right]$. Assume that ${\left(f_n\right)}_{n\in \mathbb{N}}$ is w-K-converge $a.e.$ to a function f, then ${\left(f_n\right)}_{n\in \mathbb{N}}$ converges weakly to f in $L_{E^{\prime }}^1\left[E\right]$.

Proof. 

By a general criterion for weak convergence sequence in Banach space ([17], Corollary 2) it is enough to prove that for every subsequence ${\left(f_n^{\prime }\right)}_{n\in \mathbb{N}}$ of ${\left(f_n\right)}_{n\in \mathbb{N}}$ there exist $g_n\in co\left\{f_i^{\prime }:i\ge n\right\}$ which weakly converges to f in $L_{E^{\prime }}^1\left[E\right]$. Let ${\left(f_n^{\prime }\right)}_{n\in \mathbb{N}}$ be a subsequence of ${\left(f_n\right)}_{n\in \mathbb{N}}$, by the hypothesis

$$\frac{1}{n}{\displaystyle \sum _{i=1}^n}{f}_i^{\prime }\left(\omega \right)\to f\left(\omega \right)\mathrm{weakly}\mathrm{in}{E}^{\prime }a.e.$$

so the sequence ${\left(g_n\right)}_{n\in \mathbb{N}}$ defined by $g_n=\frac{1}{n+1}{\displaystyle \sum _{i=n}^{2n}}{f}_i^{\prime }\in co\left\{f_i^{\prime }:i\ge n\right\}$ and

$$g_n=\frac{2n}{n+1}\frac{1}{2n}{\displaystyle \sum _{i=1}^{2n}}{f}_i^{\prime }-\frac{n-1}{n+1}\frac{1}{n-1}{\displaystyle \sum _{i=1}^{n-1}}{f}_i^{\prime }$$

w-converges $a.e.$ to f. On the other hand ${\left(g_n\right)}_{n\in \mathbb{N}}$ is UI in $L_{E^{\prime }}^1\left[E\right]$, hence by ([1], Theorem 4.5) it converges weakly to f in $L_{E^{\prime }}^1\left[E\right]$. □

The next result is a different version of Theorem 1, which deals with the weak convergence. Recall that $\mathcal{R}wc\left(E^{\prime }\right)$ denoted the set of nonempty closed convex subsets of $E^{\prime }$, such that their intersection with any closed ball is weakly compact.

Theorem 4.

Let ${\left(f_n\right)}_{n\in \mathbb{N}}$ be a bounded sequence in $L_{E^{\prime }}^1\left[E\right]$. Suppose that there exist a $\mathcal{R}wc\left(E^{\prime }\right)$-valued multifunction Γ, such that $f_n\left(\omega \right)\in \mathsf{\Gamma }\left(\omega \right)$ for $a.e.$$\omega \in \Omega$ and for all n $\in \mathbb{N}$. Then, there exists a function f in $L_{E^{\prime }}^1\left[E\right]$ and a subsequence ${\left(g_n\right)}_{n\in \mathbb{N}}$ of ${\left(f_n\right)}_{n\in \mathbb{N}}$, such that for every subsequence ${\left(h_n\right)}_{n\in \mathbb{N}}$ of ${\left(g_n\right)}_{n\in \mathbb{N}}$ the following holds:

(j) 

${(\frac{1}{n}{\displaystyle \sum _{i=1}^n}{h}_i)}_{n\in \mathbb{N}}\mathit{w}-\mathit{converges}a.e.\mathit{to}f$;

(jj) 

${\left(1_{\left\{\parallel h_n\parallel <n\right\}}{h}_n\right)}_{n\in \mathbb{N}}$ converges $\sigma (L_{E^{\prime }}^1\left[E\right],{\left(L_{E^{\prime }}^1\left[E\right]\right)}^{\prime })$ (weakly) to f in $L_{E^{\prime }}^1\left[E\right]$ and ${(h_n-1_{\left\{\parallel h_n\parallel <n\right\}}{h}_n)}_{n\in \mathbb{N}}$ converges $a.e.$ to 0 in $E^{\prime }$.

Proof. 

(j) As ${\left(f_n\right)}_{n\in \mathbb{N}}$ is bounded in $L_{E^{\prime }}^1\left[E\right]$, by Theorem 1 (ii) there is f in $L_{E^{\prime }}^1\left[E\right]$ and a subsequence ${\left(g_n\right)}_{n\in \mathbb{N}}$ of ${\left(f_n\right)}_{n\in \mathbb{N}}$, such that

$${\left(g_n\right)}_{n\in \mathbb{N}}\mathrm{w}*-\mathrm{K}-\mathrm{converges}a.e.\mathrm{to}f.$$

(11)

Applying Komlós theorem to $(\parallel g_n(.){\parallel )}_{n\in \mathbb{N}}$ and by extracting a subsequence if necessary, we may suppose that there exists a real integrable function $\phi$, such that

$$(\parallel g_n(.){\parallel )}_{n\in \mathbb{N}}\mathrm{K}-\mathrm{converges}a.e.\mathrm{to}\phi .$$

(12)

Let ${\left(h_n\right)}_{n\in \mathbb{N}}$ be a fixed subsequence of ${\left(g_n\right)}_{n\in \mathbb{N}}$ and set $S_n:=\frac{1}{n}{\displaystyle \sum _{i=1}^n}{h}_i$. There exists by (12) $\mathcal{N}\in \mathcal{F}$ with $\mu \left(\mathcal{N}\right)=0$, such that for all $\omega \in \Omega \backslash \mathcal{N}$

$$\parallel S_n\left(\omega \right)\parallel \le \frac{1}{n}{\displaystyle \sum _{i=1}^n}\parallel h_i\left(\omega \right)\parallel \to \phi \left(\omega \right),$$

hence ${\left(S_n\left(\omega \right)\right)}_{n\in \mathbb{N}}$ is bounded and $S_n\left(\omega \right)\in K\left(\omega \right)$ where

$$K\left(\omega \right)=\mathsf{\Gamma }\left(\omega \right)\cap (\underset{n}{sup}\parallel S_n\left(\omega \right)\parallel ){\overline{B}}_{E^{\prime }}$$

is convex weakly compact in $E^{\prime }$ since $\mathsf{\Gamma }$ is $\mathcal{R}wc\left(E^{\prime }\right)$-valued. By (11) there exists ${\mathcal{N}}^{\prime }\in \mathcal{F}$ with $\mu \left({\mathcal{N}}^{\prime }\right)=0$, such that for all $\omega \in \Omega \backslash {\mathcal{N}}^{\prime }$, ${\left(S_n\left(\omega \right)\right)}_{n\in \mathbb{N}}$ w*-converges to $f\left(\omega \right)$. Hence, for all $\omega \in \Omega \backslash (\mathcal{N}\cup {\mathcal{N}}^{\prime })$, every w-convergent subsequence of ${\left(S_n\left(\omega \right)\right)}_{n\in \mathbb{N}}$ converges to $f\left(\omega \right)$. As ${\left(S_n\left(\omega \right)\right)}_{n\in \mathbb{N}}$ is w-relatively compact in $E^{\prime }$, we conclude that ${\left(S_n\left(\omega \right)\right)}_{n\in \mathbb{N}}$ w-converges to $f\left(\omega \right)$.

(jj) Applying Lemma 2 to the bounded sequence ${\left(g_n\right)}_{n\in \mathbb{N}}$, yields the existence of a subsequence ${\left(g_n^{\prime }\right)}_{n\in \mathbb{N}}$ of ${\left(g_n\right)}_{n\in \mathbb{N}}$, such that

$${\left(1_{\left\{\parallel h_n\parallel <n\right\}}{h}_n\right)}_{n\in \mathbb{N}}\mathrm{is}\mathrm{UI},$$

(13)

and

$${(h_n-1_{\left\{\parallel h_n\parallel <n\right\}}{h}_n)}_{n\in \mathbb{N}}\mathrm{converge}a.e.\mathrm{to}0\mathrm{in}{E}^{\prime }$$

(14)

for every further subsequence ${\left(h_n\right)}_{n\in \mathbb{N}}$ of ${\left(g_n^{\prime }\right)}_{n\in \mathbb{N}}$. Let ${\left(h_n\right)}_{n\in \mathbb{N}}$ be a fixed subsequence of ${\left(g_n^{\prime }\right)}_{n\in \mathbb{N}}$, we will show that ${\left(1_{\left\{\parallel h_n\parallel <n\right\}}{h}_n\right)}_{n\in \mathbb{N}}$ converges weakly to f in $L_{E^{\prime }}^1\left[E\right]$. By $\left(\mathbf{j}\right)$, the sequence ${\left(h_n\right)}_{n\in \mathbb{N}}$ w-K-converges $a.e.$ to f, and by (14), and the decomposition $1_{\left\{\parallel h_n\parallel <n\right\}}{h}_n=h_n-(h_n-1_{\left\{\parallel h_n\parallel <n\right\}}{h}_n)$ we can see that ${\left(1_{\left\{\parallel h_n\parallel <n\right\}}{h}_n\right)}_{n\in \mathbb{N}}$ also w-K-converges $a.e.$ to f. Now, as ${\left(1_{\left\{\parallel h_n\parallel <n\right\}}{h}_n\right)}_{n\in \mathbb{N}}$ is UI, by Lemma 4, ${\left(1_{\left\{\parallel h_n\parallel <n\right\}}{h}_n\right)}_{n\in \mathbb{N}}$ converges weakly to f in $L_{E^{\prime }}^1\left[E\right]$. Finally, take ${\left(g_n^{\prime }\right)}_{n\in \mathbb{N}}$ instead of ${\left(g_n\right)}_{n\in \mathbb{N}}$ in $\left(\mathbf{j}\right)$, then ${\left(g_n^{\prime }\right)}_{n\in \mathbb{N}}$ and f satisfy (j) and (jj). □

We finish this work with the following result (compare with Proposition 5.1 in [1]).

Corollary 1.

Suppose that Γ is a $\mathcal{R}wc\left(E^{\prime }\right)$-valued multifunction on Ω and H is a UI set in $L_{E^{\prime }}^1\left[E\right]$, such that $f\left(\omega \right)\in \mathsf{\Gamma }\left(\omega \right)$ for $a.e.$$\omega \in \Omega$ and for all $f\in H$, then H is relatively weakly compact in $L_{E^{\prime }}^1\left[E\right]$.

Proof. 

By Eberlein–Smulian’s theorem, the conclusion to be derived is equivalent with H being sequentially relatively weakly compact. Let ${\left(f_n\right)}_{n\in \mathbb{N}}$ be a bounded sequence in H. Since $f_n\left(\omega \right)\in \mathsf{\Gamma }\left(\omega \right)$ for $a.e.$$\omega \in \Omega$ and for all $n\in \mathbb{N}$, by Theorem 3 (jj) there is a function f in $L_{E^{\prime }}^1\left[E\right]$ and a subsequence ${\left(h_n\right)}_{n\in \mathbb{N}}$ of ${\left(f_n\right)}_{n\in \mathbb{N}}$, such that ${\left(1_{\left\{\parallel h_n\parallel <n\right\}}{h}_n\right)}_{n\in \mathbb{N}}$ converges weakly to f in $L_{E^{\prime }}^1\left[E\right]$, and ${\left(1_{\left\{\parallel h_n\parallel \ge n\right\}}{h}_n\right)}_{n\in \mathbb{N}}$ converges $a.e.$ to 0 in $E^{\prime }$. On the other hand, since ${\left(h_n\right)}_{n\in \mathbb{N}}$ is UI, ${\left(1_{\left\{\parallel h_n\parallel \ge n\right\}}{h}_n\right)}_{n\in \mathbb{N}}$ converges strongly to 0 in $L_{E^{\prime }}^1\left[E\right]$, and then ${\left(h_n\right)}_{n\in \mathbb{N}}$ converges weakly to f in $L_{E^{\prime }}^1\left[E\right]$. Hence, H is sequentially relatively weakly compact in $L_{E^{\prime }}^1\left[E\right]$. □