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|
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.