Regular conditional probability

Regular conditional probability is a concept that has developed to overcome certain difficulties in formally defining conditional probabilities for continuous probability distributions.

Motivation

Normally we define the conditional probability of an event A given an event B as:

${\mathfrak {P}}(A|B)={\frac {{\mathfrak {P}}(A\cap B)}{{\mathfrak {P}}(B)}}.$

The difficulty with this arises when the event B is too small to have a non-zero probability. For example, suppose we have a random variable X with a uniform distribution on $[0,1],$ and B is the event that $X=2/3.$ Clearly the probability of B in this case is ${\mathfrak {P}}(B)=0,$ but nonetheless we would still like to assign meaning to a conditional probability such as ${\mathfrak {P}}(A|X=2/3).$ To do so rigorously requires the definiton of a regular conditional probability.

Definition

Let $(\Omega ,{\mathcal {F}},{\mathfrak {P}})$ be a probability space, and let $T:\Omega \rightarrow E$ be a random variable, defined as a measurable function from $\Omega$ to its state space $(E,{\mathcal {E}}).$ Then a regular conditional probability is defined as a function $\nu :E\times {\mathcal {F}}\rightarrow [0,1],$ called a "transition probability", where $\nu (x,A)$ is a valid probability measure (in its second argument) on ${\mathcal {F}}$ for all $x\in E$ and a measurable function in E (in its first argument) for all $A\in {\mathcal {F}},$ such that for all $A\in {\mathcal {F}}$ and all $B\in {\mathcal {E}}$

${\mathfrak {P}}{\big (}A\cap T^{-1}(B){\big )}=\int _{B}\nu (x,A)\,d{\mathfrak {P}}{\big (}T^{-1}(x){\big )}.$

To express this in our more familiar notation:

${\mathfrak {P}}(A|T=x)=\nu (x,A).$