probability conditioning on a sigma algebra

Let $(\mathrm{\Omega },𝔅,\mu )$ be a probability space and $B\in 𝔅$ an event. Let $𝔇$ be a sub sigma algebra of $𝔅$. The of $B$ given $\mathrm{D}$ is defined to be the conditional expectation of the random variable $1_B$ defined on $\mathrm{\Omega }$, given $𝔇$. We denote this conditional probability by $\mu (B|𝔇):=E(1_B|𝔇)$. $1_B$ 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 $\mathrm{\Omega }$. The conditional probability of $B$ given $X$ is defined to be $\mu (B|{𝔅}_X)$, where ${𝔅}_X$ is the sub sigma algebra of $𝔅$, generated by (http://planetmath.org/MathcalFMeasurableFunction) $X$. The conditional probability of $B$ given $X$ is simply written $\mu (B|X)$.

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