Introduction

1. $P_B$ is clearly non-negative;
2. $P_B(\Omega)=\displaystyle{\frac{P(\Omega\cap B)}{P(B)}=\frac{P(B)}{P(B)}=1}$ ;
3. $P_B$ is countably additive: for if $\lbrace A_1,A_2,\ldots\rbrace$ is a countable collection of pairwise disjoint events in $\mathcal{F}$ , then $$P_B(\bigcup_{i=1}^{\infty} A_i)=\frac{P\big(B\cap (\bigcup A_i)\big)}{P(B)} =\frac{P\big(\bigcup (B\cap A_i)\big)}{P(B)} = \frac{\sum P(B\cap A_i)}{P(B)} = \sum_{i=1}^{\infty}P_B(A_i),$$ as $\lbrace B\cap A_1,B\cap A_2,\ldots\rbrace$ is a collection of pairwise disjoint events also.

Regular Conditional Probability

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

1. for each event $A\in\mathcal{D}$ , $P_{\mathcal{G}}(A,\cdot):\Omega\to [0,1]$ is a conditional probability (as a random variable) of $A$ given $\mathcal{G}$ ; that is,
   1. $P_{\mathcal{G}}(A,\cdot)$ is $\mathcal{G}$ -measurable and
   2. for every $B\in\mathcal{G}$ , we have $\displaystyle \int_B P_{\mathcal{G}}(A,\cdot) dP =P(A\cap B).$
2. for every outcome $\omega\in \Omega$ , $P_{\mathcal{G}}(\cdot,\omega): \mathcal{G}\to [0,1]$ 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 $\Omega$ , then by condition 2 of the definition, we can define a ``conditional'' probability measure on $\Omega$ for each outcome in the sense of the first two paragraphs.