I learnt Fatou's lemma a while ago. I am able to prove it and use it. I know examples showing that the inequality may be strict. But I don't really have an intuitive way to understand it. Any good thoughts?
[Math] Intuitively understanding Fatou’s lemma
intuitionlebesgue-integralreal-analysis
Best Answer
Since the Lebesgue integral for nonnegative functions is built up "from below" by taking suprema of "obvious" integrals, the monotone convergence theorem has always seemed to me to be the most natural of the big three (MCT, FL, LDCT). And FL is a direct corollary of the MCT: Start with the obvious, i.e.,
$$\int \inf \{f_n,f_{n+1}, \dots \} \le \int f_n.$$
From that we get
$$\lim_{n\to \infty} \int \inf \{f_n,f_{n+1}, \dots \} \le \liminf_{n\to \infty} \int f_n.$$
Really, that should be $\liminf$ on the left, but since the integrands increase, so do the integrals, so the limit exists and we're fine. Now by MCT, that limit can be moved through the integral sign, and then you have FL.