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I have to do the following problem:

Let $X$ be a random variable in $\mathcal{L}^{1}(\Omega,A,\mathbb P)$. Let $(A_n)_{n\geq 0}$ be a sequence of events in $A$ such that $\mathbb P(A_{N})\xrightarrow[n\rightarrow\infty]{}0$. Prove that $\mathbb E(X\mathbb{1}_{A_n})\xrightarrow[n\rightarrow\infty]{}0$.

I'd appreciate any help.

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I would write the expectation in integral form to better see the result:

$$E(X \cdot \mathbb{1}_{A_n}) = \int_{\Omega} X \cdot \mathbb{1}_{A_n} \,\mathrm{d}P = \int_{A_n} X \,\mathrm{d}P$$

Note that $X \in \mathcal{L}_1$ gives you that $\int_{\Omega} X \,\mathrm{d}P$ is finite. Then apply the condition $P(A_n) \to 0$.

Hint: Use the dominated convergence theorem.

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Using Fubini-Tonelli's theorem, you can write $$|E(X 1_{A_n})| \leq E(|X|1_{A_n}) = E(1_{A_n} \int _0^\infty1_{\{|X| > t\}}dt) = \int_0^\infty P(\{|X| > t\}\cap A_n)\,dt$$

and then use the dominated convergence theorem since :

  1. $P(\{|X| > t\}\cap A_n) \leq P(A_n) \to 0$

  2. $P(\{|X| > t\}\cap A_n) \leq P(|X| > t)$ and $\int_0^\infty P(|X| > t)\,dt = E(|X|) < \infty$

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Threat the cases

  1. $X=\chi_S$, where $S$ is a measurable set;
  2. then when $X$ is a linear combination of such function;
  3. then when $X\geqslant 0$ is any measurable and integrable function;
  4. the general case.
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