There are some part of this proof I couldn't understand. First, let me state the theorem and its proof which quoted from an analysis book.

EGOROFF's THEOREM :

$\displaystyle m(A)< \infty $ where $\displaystyle A\subset$\mathbb{R}$ $. Let $\displaystyle f_n $ and $\displaystyle f $ be functions defined on A such that each $\displaystyle f_n \rightarrow f$ almost everywhere on A. Then given any $\displaystyle \epsilon > 0 $ there exists a measurable subset $\displaystyle A_\epsilon \subset A$ such that $\displaystyle m(A\backslash A_\epsilon)<\epsilon $ and $\displaystyle f_n \rightarrow f $ uniformly on $\displaystyle A_\epsilon$ .

PROOF :

Given that $\displaystyle \varepsilon $>0 and positive integers m and n, define the set

$\displaystyle E_{m,n} = \bigcap_{i\geq n}$ { $\displaystyle x : |f_i (x)-f(x)|< \frac{1}{m}$ } where $\displaystyle i\in $\mathbb{N}$$ .

If U is the subset on which $\displaystyle f_n \rightarrow f $ , then for any m , $\displaystyle U\subset \bigcup_{n\geq 1}E_{m,n} \subset A $. And Hence we get $\displaystyle m(\bigcup_{n\geq 1} E_{m,n})=m(A) $ .

But, $\displaystyle E_{m,n} \subset E_{m,n+1} $ for all m and n, so $\displaystyle lim_{n\rightarrow \infty} m(A\backslash E_{m,n})$ = $\displaystyle lim_{n\rightarrow\infty}[m(A)-m(E_{m,n})] $=$\displaystyle m(A)-m(\bigcup_{n\geq1}E_{m,n})$ = 0

Thus for each m, there exists an integer $\displaystyle n_m$ such that $\displaystyle m(A\backslash E_{m,n_m}) $ < $\displaystyle \frac{\varepsilon}{2^m} $.

Now let $\displaystyle A_\epsilon$ = $\displaystyle \bigcap_{m\geq1}E_{m,n_m} $. Then $\displaystyle A_\epsilon$ is measurable and $\displaystyle m(A\backslash A_\epsilon)$=$\displaystyle m(\bigcup_{m\geq1}(A\backslash E_{m,n_m})) \leq \sum_{m\geq1}m(A-E_{m,n_m})\lneq \sum_{m\geq1}\frac{\epsilon}{2^m}$=$\displaystyle \varepsilon$.

It is easy to show that $\displaystyle f_n \rightarrow f $ uniformly on A.

So, MY QUESTIONS ARE :

1) What is this set $\displaystyle E_{m,n} = \bigcap_{i\geq n}$ { $\displaystyle x : |f_i (x)-f(x)|< \frac{1}{m}$ } defined at the beginning of this proof and why it is defined like this?

2) How can we conclude that " for each m, there exists an integer $\displaystyle n_m$ such that $\displaystyle m(A\backslash E_{m,n_m}) $ < $\displaystyle \frac{\varepsilon}{2^m} $ "

3) Where does the properties m(A) < $\displaystyle \infty$ used in the proof?