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 of measurable functions such that $f_{n}(x)\to f(x)$ a.e. x, as n goes to infinity. If $|f_{n}(x)|\leq g(x)$, where g is integrable, then

$\int |f_{n}-f|\to 0$

and consequently

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

Lemma

Suppose $f$ is integrable in $R$. Then for every $\epsilon >0$, there exist a set of finite measure B such that

$\int _{B^{c}}|f|<\epsilon$

• Proof of lemma: By replacing $f$ with $|f|$ we may assume without loss of generality that $f\geq 0$.Let $B_{N}$ denote the ball of radius N centered at origin, and note that if $f_{N}(x)=f(x)\mathbf {1} _{B_{N}}(x)$, then $f_{N}\geq 0ismeasurable$, $f_{N}(x)\leq f_{N+1}(x)$, and $f_{N}(x)\to f(x)$. By the monotone convergenc theorem, we must have