Let $(X,\mathcal{F})$ and $(Y,\mathcal{G})$ be \reftext{def:sigma-algebra-measurable-space-2026a}{measurable spaces}. A function $f:X\to Y$ is \textbf{measurable} (with respect to $\mathcal{F}$ and $\mathcal{G}$) if $f^{-1}(B)\in\mathcal{F}$ for every $B\in\mathcal{G}$.

A real-valued function $f:X\to\mathbb{R}$ is called \textbf{measurable} if it is measurable with respect to $\mathcal{F}$ and the \reftext{def:borel-sigma-algebra-real-line-2026a}{Borel $\sigma$-algebra} $\mathcal{B}(\mathbb{R})$.

\textbf{Generator criterion.} If $\mathcal{G}=\sigma(\mathcal{C})$ is \reftext{def:generated-sigma-algebra-2026a}{generated} by a family $\mathcal{C}$, then $f$ is measurable as soon as $f^{-1}(C)\in\mathcal{F}$ for every $C\in\mathcal{C}$: the family $\{B\subseteq Y: f^{-1}(B)\in\mathcal{F}\}$ is a $\sigma$-algebra on $Y$, because taking preimages commutes with complements and countable unions; it contains $\mathcal{C}$, hence contains $\sigma(\mathcal{C})$. In particular, since every open subset of $\mathbb{R}$ is a countable union of open intervals with rational endpoints and the intervals $(a,\infty)$ generate the intervals via countable intersections and complements, $f:X\to\mathbb{R}$ is measurable if and only if

for every $a\in\mathbb{R}$.