Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space, let $(E, \mathcal{B}(E))$ be a Polish space equipped with its Borel $\sigma$-algebra, and let $X: \Omega \to E$ be a random variable. Let $\mathcal{G} \subset \mathcal{F}$ be a sub-$\sigma$-algebra. Then there exists a **regular conditional distribution** of $X$ given $\mathcal{G}$: a map \begin{align*} \mu: \Omega \times \mathcal{B}(E) &\to [0,1] \end{align*} such that: 1. For each $\omega \in \Omega$, $\mu(\omega, \cdot)$ is a Borel probability measure on $E$. 2. For each $B \in \mathcal{B}(E)$, $\mu(\cdot, B)$ is $\mathcal{G}$-measurable. 3. For each $B \in \mathcal{B}(E)$ and $A \in \mathcal{G}$: \begin{align*} \mathbb{P}(X \in B, \, A) = \int_A \mu(\omega, B) \, d\mathbb{P}(\omega). \end{align*}