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Monotone convergence theorem

For an increasing sequence of nonnegative measurable functions, the integral of the limit equals the limit of the integrals.

Monotone convergence theorem

Monotone Convergence Theorem (Beppo Levi): Let $(X,\Sigma,\mu)$ be a measure space and let $(f_n)_{n\ge 1}$ be a sequence of nonnegative measurable functions $f_n:X\to[0,\infty]$ such that $f_n(x)\le f_{n+1}(x)$ for all $x\in X$ and all $n$ . Define $f(x)=\lim_{n\to\infty} f_n(x)$ (possibly $+\infty$ ). Then

\int_X f\,d\mu \;=\; \lim_{n\to\infty}\int_X f_n\,d\mu,

where the integrals are the (possibly infinite) Lebesgue integrals of nonnegative functions .

If the monotone increase and the pointwise limit hold only almost everywhere , the conclusion is unchanged because modifying functions on a null set does not affect their integral (see a.e. equality ). Along with Fatou's lemma and the dominated convergence theorem , it is one of the main tools for exchanging limits and Lebesgue integration .