The Product of Two Probability Kernels is a Probability Kernel

In the setting of Measure Spaces and the Lebesgue Integral: Standing Notation, let $(W,\mathcal{W})$, $(Y,\mathcal{Y})$ and $(Z,\mathcal{Z})$ be measurable spaces, let $\alpha$ be a probability kernel from $(W,\mathcal{W})$ to $(Y,\mathcal{Y})$ and $\beta$ a probability kernel from $(W,\mathcal{W})$ to $(Z,\mathcal{Z})$, and write $\alpha_{w}=\alpha(w,\cdot)$ and $\beta_{w}=\beta(w,\cdot)$. Let $\mathcal{Y}\otimes\mathcal{Z}$ be the product $\sigma$-algebra. For $w\in W$ the probability measures $\alpha_{w}$ and $\beta_{w}$ are finite, hence $\sigma$-finite, so their product measure $\alpha_{w}\otimes\beta_{w}$ on $\mathcal{Y}\otimes\mathcal{Z}$ is defined; it is a probability measure, since its value at $Y\times Z$ is $\alpha_{w}(Y)\,\beta_{w}(Z)=1$ by the defining property of the product measure, so its values lie in $[0,1]$ by Basic Properties of a Measure §monotone.

1. (The product kernel)¶ The function $\alpha\otimes\beta:W\times(\mathcal{Y}\otimes\mathcal{Z})\to\mathbb{R}$, $(\alpha\otimes\beta)(w,E)=(\alpha_{w}\otimes\beta_{w})(E)$, is a probability kernel from $(W,\mathcal{W})$ to $(Y\times Z,\mathcal{Y}\otimes\mathcal{Z})$.