Let $(X,\mathcal{F},\mu)$ be a \reftext{def:measure-measure-space-2026a}{measure space}, and let $(f_m)_{m\in\mathbb{N}}$ be a \reftext{def:sequence-in-set-2026a}{sequence} of \reftext{def:measurable-function-2026a}{measurable} functions $f_m:X\to\mathbb{R}$ such that for every $x\in X$ the sequence $(f_m(x))_m$ \reftext{def:limit-sequence-real-c54-2026a}{converges} to $f(x)$, for a function $f:X\to\mathbb{R}$. Suppose there is an \reftext{def:lebesgue-integral-integrable-2026a}{integrable} function $g:X\to\mathbb{R}$ with $|f_m(x)|\le g(x)$ for every $x\in X$ and every $m$. Then:

1. $f$ is measurable and integrable;
2. $\int_X|f_m-f|\,d\mu\to 0$ as $m\to\infty$;
3. consequently $\int_X f_m\,d\mu\to\int_X f\,d\mu$.