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Let $\{X_n\}_{n\geq 1}$ be a sequence of integrable random variables, i.e., $X_n \in L_1$ for all $n \geq 1$.

Suppose $X_n$ converges to a random variable $X$ almost surely. Do we have that $X \in L_1$?

I know convergence in $L_p$ and convergence almost surely do not imply each other, but I can not find any counter-example for above argument..

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Counterexamples abound; for instance you can look at $(0,1)$ with the Lebesgue measure and take $X_n(\omega)=\begin{cases} 0 & \omega<1/n \\ 1/\omega & \omega \geq 1/n \end{cases}$. If it were this simple then we would not have convergence theorems like the dominated convergence theorem, monotone convergence theorem, etc.

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    I see. For dominated convergence theorem, the Fatou's Lemma makes sure $X \in L_1$. Thanks so much! – 2017-02-23