Theorem

Vitali Convergence Theorem

Analysis

The Vitali Convergence Theorem, named for the Italian mathematician Giuseppe Vitali, is a result of real analysis and measure theory generalizing the better-known Dominated Convergence Theorem of Henri Lebesgue. It characterizes convergence of a sequence of functions in the Lp norm in terms of convergence in measure together with a condition of uniform integrability, giving a criterion for Lp convergence that does not require the single dominating function Lebesgue's theorem demands.

Facts

Statement

For a finite measure space, a sequence of functions in Lp converges to a limit in Lp if and only if the sequence converges to that limit in measure and the sequence of p-th powers of the functions has uniformly absolutely continuous integrals. 1

Classification

Statement Form

Characterization Theorem 1

Connections

Has Statement Form

Characterization

Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.

In Branch

Real Analysis

Source Vitali convergence theorem (Wikipedia)

Proved By

Giuseppe Vitali

Source Vitali convergence theorem (Wikipedia)

Sources

1. Vitali convergence theorem (Wikipedia)

Finite measure case section
the following are equivalent
In Branch: Real Analysis, Lead sentence
In real analysis and measure theory, the Vitali convergence theorem, named after the Italian mathematician Giuseppe Vitali, is a g
Proved By: Giuseppe Vitali, Lead paragraph
In real analysis and measure theory, the Vitali convergence theorem, named after the Italian mathematician Giuseppe Vitali, is a generalization of the better-known dominated

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Vitali Convergence Theorem

Has Statement Form

In Branch

Proved By