regular conditional probability

Introduction

Suppose $(\mathrm{\Omega },ℱ,P)$ is a probability space and $B\in ℱ$ be an event with $P(B)>0$. It is easy to see that $P_B:ℱ\to [0,1]$ defined by

$$P_B(A):=P(A|B),$$

the conditional probability of event $A$ given $B$, is a probability measure defined on $ℱ$, since:

1. $P_B$ is clearly non-negative;

2. $P_B(\mathrm{\Omega })={\displaystyle \frac{P(\mathrm{\Omega }\cap B)}{P(B)}}={\displaystyle \frac{P(B)}{P(B)}}=1$;

3. $P_B$ is countably additive: for if $\{A_1,A_2,\mathrm{\dots }\}$ is a countable collection of pairwise disjoint events in $ℱ$, then

   $$P_B(\bigcup _{i=1}^{\mathrm{\infty }}{A}_i)=\frac{P\left(B\cap (\bigcup A_i)\right)}{P(B)}=\frac{P\left(\bigcup (B\cap A_i)\right)}{P(B)}=\frac{\sum P(B\cap A_i)}{P(B)}=\sum _{i=1}^{\mathrm{\infty }}{P}_B(A_i),$$

   as $\{B\cap A_1,B\cap A_2,\mathrm{\dots }\}$ is a collection of pairwise disjoint events also.

Regular Conditional Probability

Can we extend the definition above to $P_{𝒢}$, where $𝒢$ is a sub sigma algebra of $ℱ$ instead of an event? First, we need to be careful what we mean by $P_{𝒢}$, since, given any event $A\in ℱ$, $P(A|𝒢)$ is not a real number. And strictly speaking, it is not even a random variable, but an equivalence class of random variables (each pair differing by a null event in $𝒢$).

With this in mind, we start with a probability measure $P$ defined on $ℱ$ and a sub sigma algebra $𝒢$ of $ℱ$. A function $P_{𝒢}:𝒢\times \mathrm{\Omega }\to [0,1]$ is a called a regular conditional probability if it has the following properties:

1. for each event $A\in 𝒢$, $P_{𝒢}(A,\cdot ):\mathrm{\Omega }\to [0,1]$ is a conditional probability (http://planetmath.org/ProbabilityConditioningOnASigmaAlgebra) (as a random variable) of $A$ given $𝒢$; that is,

   1. $P_{𝒢}(A,\cdot )$ is $𝒢$-measurable (http://planetmath.org/MathcalFMeasurableFunction) and

   2. for every $B\in 𝒢$, we have ${\displaystyle {\int }_B}{P}_{𝒢}(A,\cdot )𝑑P=P(A\cap B).$

2. for every outcome $\omega \in \mathrm{\Omega }$, $P_{𝒢}(\cdot ,\omega ):𝒢\to [0,1]$ is a probability measure.

There are probability spaces where no regular conditional probabilities can be defined. However, when a regular conditional probability function does exist on a space $\mathrm{\Omega }$, then by condition 2 of the definition, we can define a “conditional” probability measure on $\mathrm{\Omega }$ for each outcome in the sense of the first two paragraphs.