math lessons - Egorov's theorem

In mathematics, Egorov's theorem in real analysis establishes a condition for the uniform convergence of a sequence of measurable functions.

$\left \lbrace f_n \right \rbrace$

be a sequence of measurable functions that converge almost everywhere on a measurable set A to a limit function $f$ .

Then for every

ε > 0,

there exists a set

$B \subset A$

such that

m(B) < ε

and

$\left \lbrace f_n \right \rbrace$

converges to $f$ uniformly on the difference set

$A \cap B^C$ .

Egorov's theorem can be used along with compactly supported continuous functions to prove Lusin's theorem for integrable functions.

The theorem is named for Dmitri Egorov , a Russian physicist and geometer.

References

Beals, Richard (2004). Analysis: An Introduction. New York: Cambridge University Press. ISBN 0-521-60047-2.
Weisstein, Eric W., et al. (2005). Egorov's Theorem. Retrieved April 19, 2005.