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 conditional probability 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 $X$ . 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.