probability conditioning on a sigma algebra

Let $(\Omega,\mathfrak{B},\mu)$ be a probability space and $B\in\mathfrak{B}$ an event. Let $\mathfrak{D}$ be a sub sigma algebra of $\mathfrak{B}$ . The of $B$ given $\mathfrak{D}$ is defined to be the conditional expectation of the random variable $1_{B}$ defined on $\Omega$ , given $\mathfrak{D}$ . We denote this conditional probability by $\mu(B|\mathfrak{D}):=E(1_{B}|\mathfrak{D})$ . $1_{B}$ is also known as the indicator function.

Similarly, we can define a conditional probability given a random variable. Let $X$ be a random variable defined on $\Omega$ . The conditional probability of $B$ given $X$ is defined to be $\mu(B|\mathfrak{B}_{X})$ , where $\mathfrak{B}_{X}$ is the sub sigma algebra of $\mathfrak{B}$ , generated by (http://planetmath.org/MathcalFMeasurableFunction) $X$ . The conditional probability of $B$ given $X$ is simply written $\mu(B|X)$ .

Remark. Both $\mu(B|\mathfrak{D})$ and $\mu(B|X)$ are random variables, the former is $\mathfrak{D}$ -measurable, and the latter is $\mathfrak{B}_{X}$ -measurable.

Title	probability conditioning on a sigma algebra
Canonical name	ProbabilityConditioningOnASigmaAlgebra
Date of creation	2013-03-22 16:25:05
Last modified on	2013-03-22 16:25:05
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	7
Author	CWoo (3771)
Entry type	Definition
Classification	msc 60A99
Classification	msc 60A10