Theorem

Egorov's Theorem

Analysis

On a finite measure space, a sequence of measurable functions that converges pointwise almost everywhere converges uniformly on a subset whose complement has arbitrarily small measure. Named for Dmitri Egorov, it relates pointwise and uniform convergence in measure theory.

Facts

Statement

If a sequence of measurable functions converges pointwise almost everywhere on a set of finite measure, then for every positive number there is a subset of measure less than that number outside of which the sequence converges uniformly. 1

Proof Year

1911 1

Classification

Statement Form

Existence Theorem 1

Statement Form

Inequality 1

Connections

Source Egorov's Theorem (Wikipedia)

Sources

1. Egorov's Theorem (Wikipedia)

Wikimedia Foundation

Lead section, opening definition
Egorov's theorem establishes a condition for the uniform convergence of a pointwise convergent sequence of measurable functions.
Lead section
It is also named Severini-Egoroff theorem or Severini-Egorov theorem, after Carlo Severini, an Italian mathematician, and Dmitri Egorov, a Russian mathematician and geometer, who published independent proofs respectively in 1910 and 1911.
Proved By: Carlo Severini, Lead
The first proof of the theorem was given by Carlo Severini in 1910: he used the result as a tool in his research on series of orthogonal functions.

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Egorov's Theorem

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