About Fatou's lemma and Egorov's theorem My professor in real analysis said that Fatou's lemma and Egorov's theorem are "almost never useful": Fatou's lemma is just a step in proving dominated convergence theorem and egorov's theorem is used to prove bounded convergence theorem and it is unnecessary.
It seems the case when I looked up a number of real analysis texts. There are virtually no example/exercise using them.
Does somebody have some nice examples showing these two theorems can be useful? Thanks a lot.
 A: Let me give you a concrete example where Fatou's Lemma is superior when it comes to effectiveness. 
Calculate the limit $$\lim_{n\rightarrow \infty} \int_{0}^{\infty}\frac{n^{2}}{1+n^{2}x^{2}}e^{-\frac{x^{2}}{n^{3}}}\, dx$$
Now if you want to prove that the sequence of functions $\left\{f_{n}(x)\right\}_{n\geq1}$ is non-decreasing in $n$, then be my guest. But simply applying Fatou's Lemma shows that the limit is $+ \infty$ without getting your hands dirty. 
As another user mentioned, Fatou's Lemma comes in great hands whenever you have a bounded sequence in $L^{p}$ and want to prove that the pointwise limit is contained in $L^{p}$ aswell. 
Now when it comes to Egoroff's Theorem i think there is a pretty general statement about a special case of weak compactness of $L^{p}$ for $p\in (1,\infty)$, even when the measure fails to be $\sigma$-finite. More precisely, whenever we have that $\left\{f_{n}\right\}_{n\geq1}$ with $$\sup_{n\geq 1}||f_{n}||_{p} <\infty$$ and $f_{n}\rightarrow f$ pointwise a.e , then we will have that $f_{n}\rightarrow f$ weakly. 
I think Egoroff's Theorem is crucial here if I'm not misstaken.
