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 :

where . Let and be functions defined on A such that each almost everywhere on A. Then given any there exists a measurable subset such that and uniformly on .

PROOF :

Given that >0 and positive integers m and n, define the set

{ } where .

If U is the subset on which , then for any m , . And Hence we get .

But, for all m and n, so = = = 0

Thus for each m, there exists an integer such that < .

Now let = . Then is measurable and = = .

It is easy to show that uniformly on A.

So, MY QUESTIONS ARE :

1) What is this set { } 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 such that < "

3) Where does the properties m(A) < used in the proof?