Theorem

Monotone Convergence Theorem

Analysis

If a sequence of non-negative measurable functions increases pointwise to a limit function, the limit of their integrals equals the integral of the limit function. It is one of the basic tools for interchanging limits and integration in Lebesgue's theory.

Facts

Statement

The monotone convergence theorem, in its Lebesgue-integral form due to Beppo Levi, states that for a pointwise increasing sequence of non-negative measurable functions, the integral of the limit equals the limit of the integrals, so the integral and the supremum can be interchanged. 1

Proof Year

1906 1

Classification

Statement Form

Characterization Theorem 1

Connections

Source Monotone Convergence Theorem (Wikipedia)

Henri Lebesgue

Source Monotone Convergence Theorem (Wikipedia)

Sources

1. Monotone Convergence Theorem (Wikipedia)

Wikimedia Foundation

Monotone convergence for non-negative measurable functions (Beppo Levi) section
for sequences of non-negative pointwise-increasing measurable functions 0 <= f1(x) <= f2(x) <= ..., taking the integral and the supremum can be interchanged
Monotone convergence for non-negative measurable functions (Beppo Levi) section, on its origin
It is due to Beppo Levi, who in 1906 proved a slight generalization of an earlier result by Henri Lebesgue.
Lead section, statement-form reference
In particular, infinite sums of non-negative numbers converge to the supremum of the partial sums if and only if the partial sums are bounded.
Proved By: Beppo Levi, Lead
In more advanced mathematics the monotone convergence theorem usually refers to a fundamental result in measure theory due to Lebesgue and Beppo Levi
Proved By: Henri Lebesgue, Lead
In more advanced mathematics the monotone convergence theorem usually refers to a fundamental result in measure theory due to Lebesgue and Beppo Levi

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Monotone Convergence Theorem

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