Egorov's theorem

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

In a measure space, let

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

such that

m(B) < ε

and

converges to f uniformly on the difference set

.

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.