Proof of a lemma in probability I've encountered this lemma in Chung's book as an exercise: > If $\mathbb{E}|X|<\infty$ and $ \lim_{n \to \infty} \mathbb{P}\\{\Lambda_{n}\\} = 0$, then, $$\lim_{n \to \infty} \int_{\Lambda_{n}} X\,\mathrm{d}\mathbb{P} = 0 \>.$$ Could anyone provide a detailed proof? I'm wondering since $\mathbb{E}|X|<\infty$, can I use the fact $|X|<\infty \;\mathrm{a.e.}$ then $\exists M \in \mathbb{R}^{+} \,\mathrm{s.t.}\, |X|<M \;\mathrm{a.e.}$ Then $\int_{\Lambda_{n}} X\,\mathrm{d}\mathbb{P} \leq M \,\mathbb{P}\\{\Lambda_{n}\\}\rightarrow 0$. And, can I use this lemma to prove that every $X \in L^{1}$ is uniformly integrable, using Thm 4.5.3 in Chung's book 'A course in probability theory'? Hence, every finite set of $\\{X_{n} \subset L^{1}\\}$ is uniformly integrable. However, why infinite set (possibly countably infinite) may not be uniformly integrable? Sorry to entangle these two questions together.

Hint: Write $\int_{\Lambda_n} X d\mathbb{P}$ as $\int X 1_{\Lambda_n} d\mathbb{P}$, and note that $|X 1_{\Lambda_n}| \le |X|$. Then a certain familiar theorem applies...