# regular conditional probability

## Introduction

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

 $P_{B}(A):=P(A|B),$

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

1. 1.

$P_{B}$ is clearly non-negative;

2. 2.

$P_{B}(\Omega)=\displaystyle{\frac{P(\Omega\cap B)}{P(B)}=\frac{P(B)}{P(B)}=1}$;

3. 3.

$P_{B}$ is countably additive: for if $\{A_{1},A_{2},\ldots\}$ 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 $\{B\cap A_{1},B\cap A_{2},\ldots\}$ is a collection of pairwise disjoint events also.

## Regular Conditional Probability

Can we extend the definition above to $P_{\mathcal{G}}$, where $\mathcal{G}$ is a sub sigma algebra of $\mathcal{F}$ instead of an event? First, we need to be careful what we mean by $P_{\mathcal{G}}$, since, given any event $A\in\mathcal{F}$, $P(A|\mathcal{G})$ 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 $\mathcal{G}$).

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. 1.

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

1. (a)

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

2. (b)

for every $B\in\mathcal{G}$, we have $\displaystyle\int_{B}P_{\mathcal{G}}(A,\cdot)dP=P(A\cap B).$

2. 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.

Title regular conditional probability RegularConditionalProbability 2013-03-22 16:25:24 2013-03-22 16:25:24 CWoo (3771) CWoo (3771) 8 CWoo (3771) Definition msc 60A99