Tonelli and Fubini Theorems

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

Sections. For $f:X\times Y\to[0,\infty]$ measurable with respect to $\mathcal{F}\otimes\mathcal{G}$ (in the sense of Lebesgue Integral of a Nonnegative Measurable Function) 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.

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 Lebesgue Integral of a Nonnegative Measurable Function with values in $[0,\infty]$.

Fubini. If $f:X\times Y\to\mathbb{R}$ is 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.