Monotone convergence theorem - Imported Knowlpedia Content

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.