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# 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 *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.

## Mathematics Subject Classification

60A99*no label found*60A10

*no label found*

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