Let $(X,\mathcal{F},\mu)$ and $(Y,\mathcal{G},\nu)$ be $\sigma$-finite \reftext{def:measure-measure-space-2026a}{measure spaces} and let $\mu\otimes\nu$ be the \reftext{thm:product-measure-2026a}{product measure} on the \reftext{def:product-sigma-algebra-2026a}{product $\sigma$-algebra}.

\textbf{Sections.} For $f:X\times Y\to[0,\infty]$ measurable with respect to $\mathcal{F}\otimes\mathcal{G}$ (in the sense of \ref{def:lebesgue-integral-nonnegative-2026a}) and $x\in X$, the section $f_x:Y\to[0,\infty]$, $f_x(y)=f(x,y)$, is measurable with respect to $\mathcal{G}$; symmetrically for sections in the other variable.

\textbf{Tonelli.} For every $\mathcal{F}\otimes\mathcal{G}$-measurable $f:X\times Y\to[0,\infty]$, the function $x\mapsto\int_Y f_x\,d\nu$ is $\mathcal{F}$-measurable, the symmetric function is $\mathcal{G}$-measurable, and

all integrals being those of \ref{def:lebesgue-integral-nonnegative-2026a} with values in $[0,\infty]$.

\textbf{Fubini.} If $f:X\times Y\to\mathbb{R}$ is \reftext{def:lebesgue-integral-integrable-2026a}{integrable} with respect to $\mu\otimes\nu$, then for every $x$ outside a set $N\in\mathcal{F}$ with $\mu(N)=0$ the section $f_x$ is integrable with respect to $\nu$; the function equal to $\int_Y f_x\,d\nu$ off $N$ and to $0$ on $N$ is integrable with respect to $\mu$; and the displayed identity of iterated integrals holds for $f$, with the symmetric statement in the other order.