Dominated Convergence Theorem

In measure theory, the dominated convergence theorem is a cornerstone of Lebesgue integration. It can be viewed as a culmination of all efforts, and is a general statement about the interplay between limits and integrals.

Statement and proof of Theorem

Statement

Suppose $\{f_{n}\}$ is a sequence in $\mathbf {L^{1}}$ such that $f_{n}(x)\to f(x)$ a.e. x, as n goes to infinity. If $|f_{n}(x)|\leq g(x)$ and $g\in \mathbf {L^{1}}$, where g is integrable, then $f\in \mathbf {L^{1}}$ and

$\int f_{n}\to \int f$