In the book Measure and Intrgral - Wheeden, Zygmund (p.57), I saw the Egorove's theorem and its proof. I puzzled with the statements of the Egorove's theorem, and a Lemma needed in the proof of Theorem, as the authors described. They proceed as follows:
(4.17) Egorove's Theorem: Suppose $\{f_k\}$ is a sequence of measurable functions which converges a.e. on a set $E$ to a finite limit $f$. Then given $\epsilon>0$, there exists a closed subset $F$ of $E$ such that $|E-F|<\epsilon$ and $\{f_k\}$ converges uniformly to $f$ on $F$.
Before proving this theorem, the authors prove following lemma.
(4.18) Lemma: With the same hypothesis as in Egorove's theorem, given $\epsilon,\eta>0$, there exists a closed subset $F$ of $E$ and an integer $K$ such that $|E-F|<\eta$ and $|f_k(x)-f(x)|<\epsilon$ for $x\in F$ and $k>K$.
Considering the statements of Lemma 4.18 and Theorem 4.17, I arrived at following question which I am unable to solve.
Question: Is it not true that statement of the Lemma exactly says that $\{f_k\}$ converges uniformly to $f$ on $E-F$? If yes, then why does the authors try to prove Egorove's theorem after proving the Lemma? If not, what is the difference between statements of Lemma and Theorem?
The difference is that in the Lemma, you have two "precisions" $\epsilon, \eta$. The first determines the "size" of the set $F$, i.e. $|E-F|<\epsilon$ and the second determines the closeness of $f_k$ to $f$, i.e. $|f_k -f|<\eta$ on $F$ for $K$ large.
Note that the set $F$ is allowed to depend on the closeness $\eta$!
In the statement of the theorem, we fix one set $F$, depending only on $\epsilon$. This set will then work for every precision $\eta$, because the statement of the theorem says that $f_k \to f$ uniformly on $F$.
Short summary: The Lemma allows the set $F$ to change with the "closeness" of $f_k$ to $f$, whereas the theorem yields one fixed set $F$, which allows arbitrary closeness (uniform convergence).