Sections of Product-Measurable Sets and Maps Are Measurable, and Insertion Maps into Products Are Measurable

Let $(X,\mathcal{F})$, $(Y,\mathcal{G})$, $(Z,\mathcal{H})$ and $(V,\mathcal{V})$ be measurable spaces, and let $\mathcal{F}\otimes\mathcal{G}$ denote the product $\sigma$-algebra on the Cartesian product $X\times Y$, with the analogous notation for other pairs. For $S\subseteq X\times Y$ and $x\in X$ the section of $S$ at $x$ is $S_x=\{y\in Y:(x,y)\in S\}$; for $y\in Y$ the section of $S$ at $y$ is $S^{y}=\{x\in X:(x,y)\in S\}$. Measurability of a map between measurable spaces is with respect to the named $\sigma$-algebras. Then:

1. (Sections of sets.) For every $S\in\mathcal{F}\otimes\mathcal{G}$, every $x\in X$ and every $y\in Y$, one has $S_x\in\mathcal{G}$ and $S^{y}\in\mathcal{F}$.

2. (Insertion maps.) For every $x\in X$ the map $\iota_x:Y\to X\times Y$, $\iota_x(y)=(x,y)$, is measurable with respect to $\mathcal{G}$ and $\mathcal{F}\otimes\mathcal{G}$; for every $y\in Y$ the map $\iota^{y}:X\to X\times Y$, $\iota^{y}(x)=(x,y)$, is measurable with respect to $\mathcal{F}$ and $\mathcal{F}\otimes\mathcal{G}$.

3. (Sections of maps.) Let $\Psi:X\times Y\to V$ be measurable with respect to $\mathcal{F}\otimes\mathcal{G}$ and $\mathcal{V}$. Then for every $x\in X$ the map $y\mapsto\Psi(x,y)$ is measurable with respect to $\mathcal{G}$ and $\mathcal{V}$, and for every $y\in Y$ the map $x\mapsto\Psi(x,y)$ is measurable with respect to $\mathcal{F}$ and $\mathcal{V}$. The same holds for a map $\Theta:X\times Y\to[0,\infty]$ satisfying $\{(x,y):\Theta(x,y)>a\}\in\mathcal{F}\otimes\mathcal{G}$ for every real $a$ (the measurability criterion of the integral of a nonnegative function), the sections $y\mapsto\Theta(x,y)$ and $x\mapsto\Theta(x,y)$ then satisfying the same criterion with respect to $\mathcal{G}$, respectively $\mathcal{F}$.

4. (Insertion into a triple product.) Identify $X\times Y\times Z$ elementwise with $(X\times Y)\times Z$, equip it with $(\mathcal{F}\otimes\mathcal{G})\otimes\mathcal{H}$, and write $\Psi(x,y,z)$ for $\Psi((x,y),z)$. For every $x\in X$ the map $(y,z)\mapsto((x,y),z)$ from $Y\times Z$ is measurable with respect to $\mathcal{G}\otimes\mathcal{H}$ and $(\mathcal{F}\otimes\mathcal{G})\otimes\mathcal{H}$; for every $y\in Y$ the map $(x,z)\mapsto((x,y),z)$ from $X\times Z$ is measurable with respect to $\mathcal{F}\otimes\mathcal{H}$ and $(\mathcal{F}\otimes\mathcal{G})\otimes\mathcal{H}$; and for every $z\in Z$ the map $(x,y)\mapsto((x,y),z)$ from $X\times Y$ is measurable with respect to $\mathcal{F}\otimes\mathcal{G}$ and $(\mathcal{F}\otimes\mathcal{G})\otimes\mathcal{H}$. Consequently, if $\Psi:X\times Y\times Z\to V$ is measurable with respect to $(\mathcal{F}\otimes\mathcal{G})\otimes\mathcal{H}$ and $\mathcal{V}$, then each of the maps $(y,z)\mapsto\Psi(x,y,z)$ (fixed $x$), $(x,z)\mapsto\Psi(x,y,z)$ (fixed $y$) and $(x,y)\mapsto\Psi(x,y,z)$ (fixed $z$) is measurable with respect to $\mathcal{G}\otimes\mathcal{H}$, $\mathcal{F}\otimes\mathcal{H}$, respectively $\mathcal{F}\otimes\mathcal{G}$, and $\mathcal{V}$.

5. (Coordinate projections and rearrangements.) The coordinate projections $(x,y)\mapsto x$ and $(x,y)\mapsto y$ on $X\times Y$ are measurable with respect to $\mathcal{F}\otimes\mathcal{G}$ and $\mathcal{F}$, respectively $\mathcal{G}$; the map $(x,y)\mapsto(y,x)$ from $X\times Y$ to $Y\times X$ is measurable with respect to $\mathcal{F}\otimes\mathcal{G}$ and $\mathcal{G}\otimes\mathcal{F}$; and the map $((x,y),z)\mapsto(z,x)$ from $(X\times Y)\times Z$ to $Z\times X$ is measurable with respect to $(\mathcal{F}\otimes\mathcal{G})\otimes\mathcal{H}$ and $\mathcal{H}\otimes\mathcal{F}$.