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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epslion>0}