# probability conditioning on a sigma algebra

Let $(\mathrm{\Omega},\U0001d505,\mu )$ be a probability space^{} and $B\in \U0001d505$ an event. Let $\U0001d507$ be a sub sigma algebra of $\U0001d505$. The * of $B$ given $\mathrm{D}$* is defined to be the conditional expectation of the random variable^{} ${1}_{B}$ defined on $\mathrm{\Omega}$, given $\U0001d507$. We denote this conditional probability^{} by $\mu (B|\U0001d507):=E({1}_{B}|\U0001d507)$. ${1}_{B}$ 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 $\mu (B|{\U0001d505}_{X})$, where ${\U0001d505}_{X}$ is the sub sigma algebra of $\U0001d505$, generated by (http://planetmath.org/MathcalFMeasurableFunction) $X$. The conditional probability of $B$ given $X$ is simply written $\mu (B|X)$.

Remark. Both $\mu (B|\U0001d507)$ and $\mu (B|X)$ are random variables, the former is $\U0001d507$-measurable, and the latter is ${\U0001d505}_{X}$-measurable.

Title | probability conditioning on a sigma algebra |
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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 |