# product measure

Let $({E}_{1},{\mathcal{B}}_{1}({E}_{1}))$ and $({E}_{2},{\mathcal{B}}_{2}({E}_{2}))$ be two measurable spaces^{}, with measures^{} ${\mu}_{1}$ and ${\mu}_{2}$. Let ${\mathcal{B}}_{1}\times {\mathcal{B}}_{2}$ be the sigma algebra on ${E}_{1}\times {E}_{2}$ generated by subsets of the form ${B}_{1}\times {B}_{2}$, where ${B}_{1}\in {\mathcal{B}}_{1}({E}_{1})$ and ${B}_{2}\in {\mathcal{B}}_{2}({E}_{2})$.

The product measure^{} ${\mu}_{1}\times {\mu}_{2}$ is defined to be the unique measure on the measurable space $({E}_{1}\times {E}_{2},{\mathcal{B}}_{1}\times {\mathcal{B}}_{2})$ satisfying the property

$${\mu}_{1}\times {\mu}_{2}({B}_{1}\times {B}_{2})={\mu}_{1}({B}_{1}){\mu}_{2}({B}_{2})\text{for all}{B}_{1}\in {\mathcal{B}}_{1}({E}_{1}),{B}_{2}\in {\mathcal{B}}_{2}({E}_{2}).$$ |

Title | product measure |
---|---|

Canonical name | ProductMeasure |

Date of creation | 2013-03-22 12:00:33 |

Last modified on | 2013-03-22 12:00:33 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 7 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 28A35 |