Disintegration of a Probability Measure on a Product with a Polish Factor: Existence and Almost-Everywhere Uniqueness of the Conditional Kernel

In the setting of Measure Spaces and the Lebesgue Integral: Standing Notation, let $(Y,\mathcal{Y})$ be a measurable space, let $(Z,d)$ be a complete and separable metric space with Borel $\sigma$-algebra $\mathcal{B}(Z)$, and let $\pi$ be a measure on the product $\sigma$-algebra $\mathcal{Y}\otimes\mathcal{B}(Z)$ with $\pi(Y\times Z)=1$. Let $\mu$ be its first marginal, $\mu(A)=\pi(A\times Z)$ for $A\in\mathcal{Y}$; it is the image measure of $\pi$ under the projection $(y,z)\mapsto y$, which is measurable with respect to $\mathcal{Y}\otimes\mathcal{B}(Z)$ and $\mathcal{Y}$ by claim 5 of Sections of Product-Measurable Sets and Maps Are Measurable, and Insertion Maps into Products Are Measurable, hence a probability measure on $(Y,\mathcal{Y})$ by claim 1 of Image Measures, Measures with Densities, and Change of Variables. For a probability kernel $\kappa$ from $(Y,\mathcal{Y})$ to $(Z,\mathcal{B}(Z))$ we write $\kappa_{y}=\kappa(y,\cdot)$, and $\mu\otimes\kappa$ is the composite measure of Integration Against a Probability Kernel: Measurable Sections, the Composite Measure on the Product and the Iterated Integral §composite.

1. (Existence)¶ There is a probability kernel $\kappa$ from $(Y,\mathcal{Y})$ to $(Z,\mathcal{B}(Z))$ with $\pi=\mu\otimes\kappa$; by Integration Against a Probability Kernel: Measurable Sections, the Composite Measure on the Product and the Iterated Integral §composite and Integration Against a Probability Kernel: Measurable Sections, the Composite Measure on the Product and the Iterated Integral §rectangles this means

$$\pi(A\times B)=\int_{Y}\mathbf{1}_{A}(y)\,\kappa(y,B)\,\mu(dy)\qquad(A\in\mathcal{Y},\ B\in\mathcal{B}(Z)).$$

2. (Uniqueness)¶ If $\kappa$ and $\kappa'$ are probability kernels from $(Y,\mathcal{Y})$ to $(Z,\mathcal{B}(Z))$ with $\pi=\mu\otimes\kappa=\mu\otimes\kappa'$, then the set of the $y\in Y$ with $\kappa_{y}\ne\kappa'_{y}$ belongs to $\mathcal{Y}$ and has $\mu$-measure $0$.