When can I interchange $\liminf$ and integral?

Question

Let $g_n:[0,T] \to \mathbb{R}$ be a sequence.

When can I say that $$\int_0^T \liminf_{n \to \infty} g_n(t) \leq \liminf_{n \to \infty}\int_0^T g_n(t)?$$ Under what circumstances?

Answer 1

See Fatou's Lemma. ${}{}{}{}{}{}$

Answer 2

you can say that :$\int_0^T \liminf_{n \to \infty} g_n(t) \leq \liminf_{n \to \infty}\int_0^T g_n(t)$ if $g_n$ is a sequence of Borel functions on $(\omega,G,t)$ , it's clear that your sequence : $g_n\geq0$ Then just to use the "Fatou’s lemma " , for information pleas see Theorem 1.1 page 1:http://pages.stat.wisc.edu/~doksum/STAT709/n709-4.pdf