Proof of the Dunford-Pettis theorem in the context of probability spaces

Question

I'd like to know if there's a proof of the Dunford-Pettis theorem without using relatively advanced theorems of functional analysis such as Eberlein–Smulian Theorem. Since I'm only interested in probability spaces, I'd like a proof that only uses standard measure theory theorems which you can find, for instance, in "Measure and Integration" (by Salamon) and "Advanced Probability" (by Sokol and Rønn-Nielsen).

Below is the version of the Dunford-Pettis theorem that I'm interest in.

Dunford-Pettis theorem: Let $(\Omega,\Sigma,\mathbb{P})$ be a probability space and $\{X_n\}_{n\in\mathbb{N}}\subseteq L^1(\mathbb{P})$ be a collection of integrable random variables. Then the following propositions are equivalent:

1. Every sequence of $\{X_n\}_{n\in\mathbb{N}}$ has a further subsequence $\{X_{n_k}\}_{k\in\mathbb{N}}$ such that exists $X\in L^1(\mathbb{P})$ with $\mathbb{E}[X_{n_k}Y]\to \mathbb{E}[XY]$ as $k\to \infty$ for all bounded random variables $Y$.
2. $\sup_{n\in\mathbb{N}}\mathbb{E}[|X_n|]<\infty$ and for all $\varepsilon >0$ there's $\delta >0$ such that for all $E\in \Sigma $ with $\mathbb{P}(E)<\delta $ we have $\sup_{n\in\mathbb{N}}\mathbb{E}[\mathbf{1}_{E}|X_n|]<\varepsilon $

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If you know a reference that contains a proof of the previous theorem which avoids relatively advanced theorems of functional analysis, please tell me. Also, if you know how to prove it, then please give me some hints because I don't know where I should start to prove that theorem.

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Thank you for your attention!

Answer 1

In some lecture notes here, pages 7, 8 and 9, a proof is given. The direction 2. $\Rightarrow $ 1. rests on extraction of a sub-sequence such that $\left(X_{n_k}\mathbf{1}_{\lvert X_{n_k}\vert \leqslant \ell}\right)$ converges weakly in $\mathbb L^2$ to some $Y_\ell$ for each fixed $\ell$. Then we can show that the sequence $\left(Y_\ell\right)_{\ell\geqslant 1}$ is Cauchy in $\mathbb L^2$ and that $(X_{n_k})$ does the job.

For the opposite implication, we can use Baire's theorem by putting a pseudo-metric on the sets by defining $\rho(A,B)=\mathbb P(A\Delta B)$.

Answer 2

You might have a look at the first volume of Probabilités et Potentiel (translated as Probability and Potentials) by C. Dellacherie and P.-A. Meyer; Dunford-Pettis is discussed in II.25.