---
tag: AAT8PVDW23
kind: definition
name: Regular conditional probability
msc: [60A10, 28A50]
defines: PHHT5W3BEJ
uses: [0NFPJVJJHX, 0SV8K0TDGY, 059TJN0NZ1, 0TTDGTEFMN, 0990W5PN13]
sources:
  - url: https://en.wikipedia.org/wiki/Regular_conditional_probability
---

## Definition

Let $(\Omega, \mathcal{F}, P)$ be a [[probability space|0NFPJVJJHX]] and let $\mathcal{G} \subseteq \mathcal{F}$ be a sub-[[σ-algebra|0SV8K0TDGY]].
A *regular conditional probability* given $\mathcal{G}$ is a map $\nu \colon \Omega \times \mathcal{F} \to [0, 1]$ such that:

1. for each $\omega \in \Omega$, the map $A \mapsto \nu(\omega, A)$ is a [[probability measure|059TJN0NZ1]] on $(\Omega, \mathcal{F})$;
2. for each $A \in \mathcal{F}$, the map $\omega \mapsto \nu(\omega, A)$ is $\mathcal{G}$-measurable;
3. for each $A \in \mathcal{F}$ and each $B \in \mathcal{G}$, $\int_B \nu(\omega, A) \, P(d\omega) = P(A \cap B)$.

## Remarks

Clauses 2 and 3 say that $\nu(\cdot, A)$ is a version of the [[conditional probability|0990W5PN13]] $P(A \mid \mathcal{G}) = E[\mathbf{1}_A \mid \mathcal{G}]$ for each $A$.
Then $E[Z \mid \mathcal{G}](\omega) = \int Z(\omega') \, \nu(\omega, d\omega')$ almost surely for every integrable $Z$.
A regular conditional probability exists when $(\Omega, \mathcal{F})$ is a standard Borel space, for example a [[Polish space|584GXXY73M]] with its Borel σ-algebra.
It is then unique up to a $P$-null set of $\omega$.
It need not exist in general.
A related notion is the regular conditional distribution of a random variable $Y$ given $X$.
It is a [[Markov kernel|2AVNDP8BJP]] $\kappa$ with $P(X \in A, Y \in B) = \int_A \kappa(x, B) \, P_X(dx)$.
It exists whenever $Y$ takes values in a standard Borel space.
Example: let $\Omega = [0, 1]^2$ with Lebesgue measure, and let $\mathcal{G}$ be generated by the first coordinate. Then $\nu((x, y), A) = \lambda(\set{y' : (x, y') \in A})$, with $\lambda$ the Lebesgue measure on $[0, 1]$, is a regular conditional probability; clause 3 is [[Fubini's theorem|0R0T1VWH5X]].
Non-example: if $\mathcal{G} = \set{\emptyset, \Omega}$ and $0 < P(A) < 1$ for some $A$, then $\nu(\omega, A) = \mathbf{1}_A(\omega)$ satisfies clause 1 but fails clause 2, since $\mathbf{1}_A$ is not constant.