Expectation of random variables with indicator function

Question

Let $(A_n)_{n \in \mathbb{N} }$ be a sequence of events such that $\lim_{n \to \infty} P(A_n) =0$. Let $X\geq 0$, $X \in \mathcal{L}^1$. Let $(X_n)_{n \in \mathbb{N} }$ be a sequence of simple non-negative random variables such that $X_n \uparrow X$. Assuming $X_0=0$, I want to show: $$0=\lim_{m \to \infty} \lim_{n \to \infty} E[X_m 1_{A_n}]$$ $$=\lim_{m \to \infty}\lim_{n \to \infty} \sum_{i=1}^{m}E[(X_i - X_{i-1})1_{A_n}]$$ I think this follows from Fatou's Lemma and dominant convergence but I don't know how to go on about it. The goal is to show that $$\lim_{n \to \infty} E[X1_{A_n}] = 0$$

Answer 1

This follows from the fact that if $Y$ is an integrable non-negative random variable, then for each sequence of sets $(A_n)_{n \in \mathbb{N} }$ such that $\lim_{n \to \infty} P(A_n) =0$, $$ \lim_{n\to +\infty}\mathbb E\left[Y\mathbf 1_{A_n}\right]=0. $$ This is easy to see when $Y$ is a linear combination of indicator functions, because such a random variable is bounded. In the general case, if $Y'$ is a linear combination of indicator functions, then $$ \limsup_{n\to +\infty}\mathbb E\left[Y\mathbf 1_{A_n}\right]\leqslant \limsup_{n\to +\infty}\mathbb E\left[\left(Y-Y'\right)\mathbf 1_{A_n}\right]\leqslant \mathbb E\left[ Y-Y' \right], $$ which can be made as small as we wish, by definition of Lebesgue integral.