Definition (Stochastic Kernel)

A stochastic kernel on some set $X$ given another set $Y$ is a function $P(\cdot\mid \cdot)$ such that

1. $P(\cdot\mid y)$ is a Probability Measure on $X$ for each fixed $y\in Y$, and;
2. $P(B\mid \cdot)$ is a Measurable Function on $Y$ for each $B\in\mathcal{B}(X)$.

The set of all stochastic kernels on $X$ given $Y$ is denoted by $\mathcal{P}(X\mid Y)$.

Definition (Disintegration)

Let  $(\mathbb{X}, \mathcal{B}(\mathbb{X}))$  and  $(\mathbb{Y}, \mathcal{B}(\mathbb{Y}))$  be Measurable Spaces, and let $\mathbb{P}$ be a Probability Measure on  $\mathbb{X} \times \mathbb{Y}$. Disintegration is the process of finding:

1. A marginal probability measure $\mu \in\mathcal{P}(\mathbb{X})$,
2. A Stochastic Kernel $\mathbb{P}(dy \mid x)\in\mathcal{P}(\mathbb{Y})$,

such that for any Measurable Function $f: \mathbb{X} \times \mathbb{Y} \to \mathbb{R}$, $$\int_{\mathbb{X} \times \mathbb{Y}} f(x, y) \mathbb{P}(dx, dy) = \int_X \left( \int_{\mathbb{Y}} f(x, y) \mathbb{P}(dy \mid x) \right) \mu(dx).$$ this expresses the joint measure $\mathbb{P}(dx, dy)$ as an integral of conditionals against the marginal measure $\mu(dx)$.