The Conditional Kernel of a Probability Measure Given a Measurable Map

In the setting of Measure Spaces and the Lebesgue Integral: Standing Notation, let $(Z,d_{Z})$ and $(Y,d_{Y})$ be metric spaces with Borel $\sigma$-algebras $\mathcal{B}(Z)$ and $\mathcal{B}(Y)$, which are $\sigma$-algebras by claim 2 of Intersections of Sigma-Algebras and Minimality of the Generated Sigma-Algebra, so that $(Z,\mathcal{B}(Z))$ and $(Y,\mathcal{B}(Y))$ are measurable spaces. Let $q:Z\to Y$ be measurable with respect to $\mathcal{B}(Z)$ and $\mathcal{B}(Y)$, let $\pi$ be a Borel measure on $(Z,d_{Z})$ with $\pi(Z)=1$, and let $\nu=q_{\#}\pi$ be its image measure $A\mapsto\pi(q^{-1}(A))$, a probability measure on $(Y,\mathcal{B}(Y))$ by claim 1 of that lemma. For $A\in\mathcal{B}(Y)$ and $B\in\mathcal{B}(Z)$ the set $B\cap q^{-1}(A)$ belongs to $\mathcal{B}(Z)$, since $q^{-1}(A)\in\mathcal{B}(Z)$ by the measurability of $q$ and a $\sigma$-algebra is closed under finite intersections (Sigma-Algebra and Measurable Space). Let $\kappa$ be a probability kernel from $(Y,\mathcal{B}(Y))$ to $(Z,\mathcal{B}(Z))$. The rectangle $E=A\times B$ belongs to the product $\sigma$-algebra $\mathcal{B}(Y)\otimes\mathcal{B}(Z)$; its section $E_{y}=\{z:(y,z)\in E\}$ is $B$ for $y\in A$ and $\varnothing$ otherwise, so $\kappa(y,E_{y})=\mathbf{1}_{A}(y)\,\kappa(y,B)$ for every $y\in Y$, because $\kappa(y,\varnothing)=0$; hence the function $y\mapsto\mathbf{1}_{A}(y)\,\kappa(y,B)$ is integrable with respect to $\nu$ by Integration Against a Probability Kernel: Measurable Sections, the Composite Measure on the Product and the Iterated Integral §sections.

(Conditional kernel)¶ A conditional kernel of $\pi$ given $q$ is a probability kernel $\kappa$ from $(Y,\mathcal{B}(Y))$ to $(Z,\mathcal{B}(Z))$ such that

$$\pi\bigl(B\cap q^{-1}(A)\bigr)=\int_{Y}\mathbf{1}_{A}(y)\,\kappa(y,B)\,\nu(dy)\qquad(A\in\mathcal{B}(Y),\ B\in\mathcal{B}(Z)),$$

where $\int_{Y}h(y)\,\nu(dy)$ denotes $\int_{Y}h\,d\nu$.