Let $(X,\mathcal{F},\mu)$ be a \reftext{def:measure-measure-space-2026a}{measure space}. A function $f:X\to[0,\infty]$ (values in the extended half-line of \ref{def:measure-measure-space-2026a}) is called \textbf{measurable} if

for every $a\in\mathbb{R}$; for real-valued $f$ this agrees with \ref{def:measurable-function-2026a} by the generator criterion stated there.

The \textbf{integral} of a measurable $f:X\to[0,\infty]$ with respect to $\mu$ is

with the integral of a nonnegative \reftext{def:simple-function-integral-2026a}{simple function} as defined there; the supremum is the \reftext{def:upper-bound-supremum-c54-2026b}{least upper bound} of the set of values when that set is bounded above, and $\infty$ otherwise. For a nonnegative simple function the two notions of integral agree, since such a function is its own largest simple minorant.