Regular Conditional Distribution

Last updated Nov 1, 2022

Let $(\Omega, \mathcal{B}, \mathbb{P})$ be a Probability Space, $X: (\Omega, \mathcal{B}) \to (S, \mathcal{S})$ be a Random Element, and $\mathcal{G} \subset \mathcal{B}$ be a Sub-Sigma Algebra. $\mu: \Omega \times \mathcal{S} \to [0,1]$ is a Regular Conditional Distribution for $X$ given $\mathcal{G}$ if

1. For each $A \in \mathcal{S}$, $\omega \mapsto \mu(\omega, A)$ is $\mathbb{E}(\mathbb{1}_{X \in A} | \mathcal{G})$ Almost Surely
2. $A \mapsto \mu(\omega, A)$ is a Probability Measure on $(S, \mathcal{S})$ for $\omega \in \Omega$ $\mathbb{P}$-Almost Surely.

1. If Space is a Standard Probability Space, then a Regular Conditional Distribution exists and is Almost Surely Unique

• Conditional Probability