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.

Theorem Statement

Consider the measure space $(X,{\mathcal {M}},\lambda )$. Suppose $\{f_{n}\}$ is a sequence in $L^{1}(\lambda )$ such that

1. $f_{n}\to f$ a.e
2. there exists $g\in L^{1}(\lambda )$ such that $|f_{n}|\leq g$ a.e. for all $n\in \mathbb {N}$

Then $f\in L^{1}(\lambda )$ and $\int f=\lim _{n\to \infty }\int f_{n}$.

Proof of Theorem

$f$ is a measurable function in the sense that it is a.e. equal to a measurable function, since it is the limit of $f_{n}$ except on a null set. Also $f\leq g$ a.e., so $f\in L^{1}(\lambda )$.