Proof of The Product of Two Probability Kernels is a Probability Kernel

Each result cited is universally quantified over the data in its own statement, and is applied below with the data named at each use.

Fix the measurable spaces $(W,\mathcal{W})$, $(Y,\mathcal{Y})$, $(Z,\mathcal{Z})$ and the probability kernels $\alpha$, $\beta$ of the statement. By the definition of a probability kernel, applied with $(W,\mathcal{W})$ and $(Y\times Z,\mathcal{Y}\otimes\mathcal{Z})$ in place of $(Y,\mathcal{Y})$ and $(Z,\mathcal{Z})$, two things must be shown about $\alpha\otimes\beta$: that for every $w\in W$ the function $(\alpha\otimes\beta)(w,\cdot)=\alpha_{w}\otimes\beta_{w}$ is a probability measure on $(Y\times Z,\mathcal{Y}\otimes\mathcal{Z})$, and that for every $E\in\mathcal{Y}\otimes\mathcal{Z}$ the function $w\mapsto(\alpha_{w}\otimes\beta_{w})(E)$ on $W$ is measurable with respect to $\mathcal{W}$ in the sense of Measurable Function and Real-Valued Measurable Function. The first was established in the preamble of the statement: for each $w\in W$, $\alpha_{w}\otimes\beta_{w}$ is a probability measure on $(Y\times Z,\mathcal{Y}\otimes\mathcal{Z})$ with values in $[0,1]$; in particular $\alpha\otimes\beta$ is indeed real-valued. The rest of the proof establishes the second.

For $E\in\mathcal{Y}\otimes\mathcal{Z}$ write $f_{E}:W\to\mathbb{R}$, $f_{E}(w)=(\alpha_{w}\otimes\beta_{w})(E)$, and let $\mathcal{D}$ be the family of those $E\in\mathcal{Y}\otimes\mathcal{Z}$ for which $f_{E}$ is measurable with respect to $\mathcal{W}$ and the Borel $\sigma$-algebra of the real line. Let $\mathcal{P}$ be the family of measurable rectangles $A\times B$ with $A\in\mathcal{Y}$ and $B\in\mathcal{Z}$, in the sense of Product Sigma-Algebra.

Step 1 ($\mathcal{P}$ is a $\pi$-system generating $\mathcal{Y}\otimes\mathcal{Z}$). Since $Y\in\mathcal{Y}$ and $Z\in\mathcal{Z}$ by property 1 of Sigma-Algebra and Measurable Space, $Y\times Z\in\mathcal{P}$, so $\mathcal{P}$ is nonempty. For $A\times B$ and $A'\times B'$ in $\mathcal{P}$ one has $(A\times B)\cap(A'\times B')=(A\cap A')\times(B\cap B')$, and $A\cap A'\in\mathcal{Y}$, $B\cap B'\in\mathcal{Z}$ because a $\sigma$-algebra is closed under finite intersections (Sigma-Algebra and Measurable Space); so $\mathcal{P}$ is closed under finite intersections and is a $\pi$-system in the sense of Dynkin's Pi-Lambda Theorem. By Product Sigma-Algebra, $\mathcal{Y}\otimes\mathcal{Z}$ is the $\sigma$-algebra generated by $\mathcal{P}$, that is, $\sigma(\mathcal{P})=\mathcal{Y}\otimes\mathcal{Z}$.

Step 2 (rectangles belong to $\mathcal{D}$). Let $A\in\mathcal{Y}$ and $B\in\mathcal{Z}$. For each $w\in W$, the defining property of the product measure in Existence and Uniqueness of the Product Measure, applied to the finite measures $\alpha_{w}$ on $(Y,\mathcal{Y})$ and $\beta_{w}$ on $(Z,\mathcal{Z})$, gives

$$f_{A\times B}(w)=(\alpha_{w}\otimes\beta_{w})(A\times B)=\alpha_{w}(A)\,\beta_{w}(B)=\alpha(w,A)\,\beta(w,B),$$

a real number. The functions $w\mapsto\alpha(w,A)$ and $w\mapsto\beta(w,B)$ are measurable with respect to $\mathcal{W}$ by the definition of a probability kernel, applied to $\alpha$ (from $(W,\mathcal{W})$ to $(Y,\mathcal{Y})$, with the set $A$) and to $\beta$ (from $(W,\mathcal{W})$ to $(Z,\mathcal{Z})$, with the set $B$). Their pointwise product $f_{A\times B}$ is therefore measurable by claim 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, applied on the measurable space $(W,\mathcal{W})$. Hence $\mathcal{P}\subseteq\mathcal{D}$.

Step 3 ($\mathcal{D}$ is a $\lambda$-system). We check the three properties of a $\lambda$-system in Dynkin's Pi-Lambda Theorem, on the set $Y\times Z$.

(1) $Y\times Z\in\mathcal{P}\subseteq\mathcal{D}$ by Steps 1 and 2.

(2) Let $E,F\in\mathcal{D}$ with $E\subseteq F$. Then $F\setminus E=F\cap\bigl((Y\times Z)\setminus E\bigr)$ belongs to $\mathcal{Y}\otimes\mathcal{Z}$ by property 2 of Sigma-Algebra and Measurable Space and the closure of a $\sigma$-algebra under finite intersections recorded there. For each $w\in W$ the measure $\alpha_{w}\otimes\beta_{w}$ is finite, so Basic Properties of a Measure §differences, applied to the measure space $(Y\times Z,\mathcal{Y}\otimes\mathcal{Z},\alpha_{w}\otimes\beta_{w})$ and the sets $E\subseteq F$, gives $(\alpha_{w}\otimes\beta_{w})(F\setminus E)=(\alpha_{w}\otimes\beta_{w})(F)-(\alpha_{w}\otimes\beta_{w})(E)$. Thus $f_{F\setminus E}=f_{F}+(-1)f_{E}$ pointwise on $W$, which is measurable by claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions on $(W,\mathcal{W})$, since $f_{E}$ and $f_{F}$ are. Hence $F\setminus E\in\mathcal{D}$.

(3) Let $(E_{m})_{m\in\mathbb{N}}$ be a sequence in $\mathcal{D}$ with $E_{m}\subseteq E_{m+1}$ for all $m$, and let $E=\bigcup_{m}E_{m}$, which belongs to $\mathcal{Y}\otimes\mathcal{Z}$ by property 3 of Sigma-Algebra and Measurable Space. Fix $w\in W$. Every $(\alpha_{w}\otimes\beta_{w})(E_{m})$ is a real number in $[0,1]$, so the set of these values is bounded above by $1$, and Basic Properties of a Measure §continuity-below, applied to the measure $\alpha_{w}\otimes\beta_{w}$ and the sequence $(E_{m})$, shows that the sequence $(f_{E_{m}}(w))_{m\in\mathbb{N}}$ converges to the real number $f_{E}(w)$. As $w$ was arbitrary, $f_{E}$ is the pointwise limit on $W$ of the measurable functions $f_{E_{m}}$, hence measurable by claim 5 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions on $(W,\mathcal{W})$. Hence $E\in\mathcal{D}$.

Step 4 (conclusion). By Steps 1 to 3, $\mathcal{P}$ is a $\pi$-system, $\mathcal{D}$ is a $\lambda$-system on $Y\times Z$, and $\mathcal{P}\subseteq\mathcal{D}$. Therefore Dynkin's Pi-Lambda Theorem, applied on the set $Y\times Z$ with this $\mathcal{P}$ and this $\mathcal{L}=\mathcal{D}$, gives $\sigma(\mathcal{P})\subseteq\mathcal{D}$, that is, $\mathcal{Y}\otimes\mathcal{Z}\subseteq\mathcal{D}$ by Step 1. So for every $E\in\mathcal{Y}\otimes\mathcal{Z}$ the function $w\mapsto(\alpha\otimes\beta)(w,E)=f_{E}(w)$ is measurable with respect to $\mathcal{W}$. Together with the fact, recalled at the outset, that each $(\alpha\otimes\beta)(w,\cdot)=\alpha_{w}\otimes\beta_{w}$ is a probability measure on $(Y\times Z,\mathcal{Y}\otimes\mathcal{Z})$, this shows by the definition of a probability kernel that $\alpha\otimes\beta$ is a probability kernel from $(W,\mathcal{W})$ to $(Y\times Z,\mathcal{Y}\otimes\mathcal{Z})$. $\square$