Egerov's Theorem : Suppose is a locally finite Borel measure and is a sequence of measurable functions defined on a measurable set with and a.e. on E.
Given we may find a closed subset such that and uniformly on 
WLOG assume for all since the set of points at which is a null set. Fix and for we define
. Since are measurable so is their difference. Then since the absolute value of a measurable function is measurable each is measurable.
Now for fixed we have that and . Therefore using continuity from below we may find a such that .
Now choose so that and define . By countable subadditivity we have that .
Fix any . We choose such that . Since if then . And by definition if then whenever . Hence uniformly on .
Finally, since is measurable, using HW5 problem 6 there exists a closed set such that . Therefore and on
Bounded Convergence Theorem : Let be a seqeunce of measurable functions bounded by , supported on a set with finite measure and a.e. Then
By assumptions on , is measurable, bounded, supported on for a.e. . Fix , then by Egerov we may find a measurable subset of such that and uniformly on . Therefore, for sufficiently large we have that for all . Putting this together yields
Since was arbitrary and is finite by assumption we are done.
Note this theorem could also be proved in one line using dominated convergence theorem with dominating function , but traditionally one proves bounded convergence before dominating convergence.
- ↑ Stein & Shakarchi, Real Analysis: Measure Theory, Integration, and Hilbert Spaces, Chapter 1 §4.3
- ↑ Stein & Shakarchi, Real Analysis: Measure Theory, Integration, and Hilbert Spaces, Chapter 2 § 1