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:lebesgue-integral-nonnegative-2026a}{measurable} functions $f_m:X\to[0,\infty]$ such that $f_m(x)\le f_{m+1}(x)$ for every $x\in X$ and every $m$. Define $f:X\to[0,\infty]$ pointwise by $f(x)=\sup_m f_m(x)$, the \reftext{def:upper-bound-supremum-c54-2026b}{least upper bound} in $[0,\infty]$ (equal to $\infty$ when the values are unbounded). Then $f$ is measurable, and

the integrals being those of \ref{def:lebesgue-integral-nonnegative-2026a}; equivalently, the nondecreasing sequence of integrals converges to $\int_X f\,d\mu$ in $[0,\infty]$.