- #1

Bashyboy

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## Homework Statement

Show that Egoroff's theorem continues to hold if the convergence is pointwise a.e. and ##f## is finite a.e.

## Homework Equations

Here is the statement of Egoroff's theorem:

Assume that ##E## has finite measure. Let ##\{f_n\}## be a sequence of measurable functions on ##E## that converges pointwise on ##E## to the real-valued function ##f##. Then for each ##\epsilon > 0##, there is a closed set ##F## contained in ##E## for which ##f_n \to f## uniformly on ##F## and ##m(E-F) < \epsilon##.

## The Attempt at a Solution

Am I allowed to use Egoroff's theorem to prove that statement?The proof of Egoroff's theorem doesn't presuppose it, so I am wondering if Egoroff's theorem is one of those theorems where the special case can actually be used to prove the general case. In fact, after the proof of Egoroff's theorem, the author writes "It is clear that Egoroff's theorem also holds if the convergence is pointwise a.e. and the limit function is finite a.e." The words "it is clear" usually indicate that the problem is easy; otherwise, this seems like it would be a pretty hard problem.

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