Let $X$ be a measure space, and let $\Phi,f_1,f_2,\dots$ be measurable functions such that $\int_X \Phi <\infty$ and $|f_n|\leq \Phi$ for each $n$ . If $f_n\rightarrow f$ almost everywhere, then $f$ is integrable and $$ \lim_{n\rightarrow\infty} \int_X f_n = \int_X f. $$

This theorem is a corollary of the Fatou-Lebesgue theorem.