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 $f_m:X\to[0,\infty]$. For $x\in X$ define

where the \reftext{def:lower-bound-infimum-c54-2026a}{infimum} and \reftext{def:upper-bound-supremum-c54-2026b}{supremum} are taken in $[0,\infty]$ with the conventions of \ref{def:measure-measure-space-2026a}, and define $\liminf_m a_m$ for a sequence $(a_m)$ in $[0,\infty]$ in the same way. Then $\liminf_m f_m$ is measurable, and