Dominated convergence theorem

Dominated convergence theorem. Let $(X,\Sigma,\mu)$ be a measure space, and let $f_n,f:X\to\mathbb{R}$ (or $\mathbb{C}$ ) be measurable functions such that $f_n\to f$ almost everywhere. Suppose there is a nonnegative $L^1$ function $g$ such that $|f_n|\le g$ almost everywhere for every $n$ . Then $f$ is Lebesgue integrable and

\lim_{n\to\infty}\int_X f_n\,d\mu \;=\; \int_X f\,d\mu, \qquad\text{and}\qquad \lim_{n\to\infty}\int_X |f_n-f|\,d\mu \;=\; 0.

In particular, $f_n\to f$ in $L^1(\mu)$ (see $L^p$ convergence with $p=1$ ).

Together with monotone convergence and Fatou's lemma, this theorem is a core tool for interchanging limits with the Lebesgue integral. It is especially useful when pointwise convergence is available but uniform bounds are only in the integrable sense.