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 $\mu(dx)$.