regular conditional probability
Introduction
Suppose $(\mathrm{\Omega},\mathcal{F},P)$ 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
$${P}_{B}(A):=P(AB),$$ 
the conditional probability^{} of event $A$ given $B$, is a probability measure defined on $\mathcal{F}$, since:

1.
${P}_{B}$ is clearly nonnegative;

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 $\mathcal{F}$, 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 ${P}_{\mathcal{G}}$, 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 ${P}_{\mathcal{G}}:\mathcal{G}\times \mathrm{\Omega}\to [0,1]$ is a called a regular conditional probability if it has the following properties:

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

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

(b)
for every $B\in \mathcal{G}$, we have ${\int}_{B}}{P}_{\mathcal{G}}(A,\cdot )\mathit{d}P=P(A\cap B).$

(a)

2.
for every outcome $\omega \in \mathrm{\Omega}$, ${P}_{\mathcal{G}}(\cdot ,\omega ):\mathcal{G}\to [0,1]$ 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 $\mathrm{\Omega}$, 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.
Title  regular conditional probability 

Canonical name  RegularConditionalProbability 
Date of creation  20130322 16:25:24 
Last modified on  20130322 16:25:24 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  8 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 60A99 