Let $X$ and $Y$ be Polish spaces, and let $\mu$ be a Borel probability measure on $X \times Y$. Let $\pi_X: X \times Y \to X$ denote the projection onto the first coordinate, and let $\nu = (\pi_X)_\# \mu$ be the pushforward (marginal) measure on $X$. Then there exists a $\nu$-almost everywhere uniquely determined family of Borel probability measures $\{\mu_x\}_{x \in X}$ on $Y$ such that: 1. For each Borel set $B \in \mathcal{B}(Y)$, the map $x \mapsto \mu_x(B)$ is Borel measurable. 2. For every bounded Borel function $g: X \times Y \to \mathbb{R}$: \begin{align*} \int_{X \times Y} g(x,y) \, d\mu(x,y) = \int_X \left(\int_Y g(x,y) \, d\mu_x(y)\right) d\nu(x). \end{align*} The measures $\mu_x$ are called the **conditional measures** or **disintegration** of $\mu$ with respect to $\pi_X$.