Disintegration of a joint law on standard borel spaces

On $(E\times T,\mathcal{S}\otimes \mathcal{T},\lambda )$ use the coordinate maps X and Y. Both targets are nonempty because the product carries a probability. By [F1], X has an RCD L given $\sigma (Y)$; by [F2] it factors as $K(Y,\cdot )$ for a probability kernel K. The RCD identity gives $\lambda (A\times B)=\int {\mathbf{1}}_B(Y)K(Y,A)d\lambda$. We prove the needed marginal substitution locally. For $g={\mathbf{1}}_C$, $\int g(Y)d\lambda =\lambda (Y^{-1}C)=\beta (C)=\int gd\beta \mathrm{.}$ Finite additivity from [F4] gives this for nonnegative simple g; apply [F6] and the local monotone convergence [F4] to get it for every nonnegative measurable g. Taking $g(y)={\mathbf{1}}_B(y)K(y,A)$ yields the rectangle formula.

We also prove kernel-integral measurability locally. Let $\mathcal{M}$ be the product-measurable sets C for which every section $C_y$ belongs to $\mathcal{S}$ and $y\mapsto K(y,C_y)$ is measurable. It contains rectangles: each section is A or empty and its evaluation is ${\mathbf{1}}_B(y)K(y,A)$. It contains the whole product. Under complements, $(C^c{)}_y=(C_y{)}^c$ is measurable and its evaluation is $1-K(y,C_y)$. Under pairwise disjoint countable unions, the sections are measurable disjoint unions, while sectionwise countable additivity gives the evaluation as a pointwise limit of measurable partial sums, measurable by [F7]. Thus [F5] gives both properties for every product event. Let $\mathcal{D}$ be the product events satisfying the iterated indicator identity. It contains rectangles by step 1.1 and the whole product by normalization. Complements subtract from the finite total one. For disjoint $C_j\in \mathcal{D}$, sectionwise countable additivity and monotone convergence from [F4], first under $K(y,\cdot )$ and then under $\beta$, prove the union identity. Hence [F5] gives every product event.

Finite nonnegative combinations of step 2.1 give both measurability and the iterated formula for nonnegative simple f. For arbitrary nonnegative f, use the prescribed $s_n\uparrow f$ of [F6]. Apply the locally proved monotone convergence [F4] under $\lambda$, in every probability section K, and then under $\beta$. By [F7] the inner limit is measurable. These passages give the displayed formula, including value infinity, with no subtraction.

If K and J both satisfy the rectangle formula, their compositions with Y are RCDs of X given $\sigma (Y)$: the collection $\{Y^{-1}(B):B\in \mathcal{T}\}$ is already a sigma-algebra. By [F3], they agree as measures outside one $\lambda$-null set. Fix the countable determining algebra $\mathcal{A}$ supplied by [F1] and set $D={\bigcup }_{A\in \mathcal{A}}\{y:K(y,A)\mathrm{\ne }J(y,A)\}$. This set is measurable. Its preimage under Y lies in the exceptional set from [F3], so $\beta (D)=\lambda (Y^{-1}D)=0$. Off D the probabilities agree on $\mathcal{A}$, hence on every event by its locally proved determining property. If the joint law comes from actual X,Y, the rectangle calculation gives the same conditional-law assertion.