Suppose that $X_1,X_2, \dots$ is a sequence of integrable random variables, which decreases to a random variable $X$. I am trying to prove that if $$\inf_{n\to\infty} \mathbb{E}(X_n) > -\infty$$ then $X \in L^1$ and $X_n$ converges to $X$ in $L^1$.

But I couldn't proceed into any way, can someone give me a hint?

  • 3
    $\begingroup$ Hint: Apply Monotone convergence theorem to $\{-X_n\}$. $\endgroup$ – Jacky Chong Oct 29 '16 at 9:20

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