Probability Kernels Between Measurable Spaces

In the setting of The Real Numbers: Standing Notation and Background, let $(Y,\mathcal{Y})$ and $(Z,\mathcal{Z})$ be measurable spaces.

(Probability kernel)¶ A probability kernel from $(Y,\mathcal{Y})$ to $(Z,\mathcal{Z})$ is a function $\kappa:Y\times\mathcal{Z}\to\mathbb{R}$ such that, for every $y\in Y$, the function $\kappa_{y}=\kappa(y,\cdot):\mathcal{Z}\to\mathbb{R}$ is a probability measure on $(Z,\mathcal{Z})$, its values lying in $[0,1]$ by Basic Properties of a Measure §monotone, and, for every $B\in\mathcal{Z}$, the function $\kappa(\cdot,B):Y\to\mathbb{R}$ is measurable with respect to $\mathcal{Y}$.