# product $\sigma $-algebra

Given measurable spaces^{} $(E,\mathcal{F})$ and $(F,\mathcal{G})$, the *product ^{} $\sigma $-algebra* $\mathcal{F}\times \mathcal{G}$ is defined to be the $\sigma $-algebra on the Cartesian product $E\times F$ generated by sets of the form $A\times B$ for $A\in \mathcal{F}$ and $B\in \mathcal{G}$.

$$\mathcal{F}\times \mathcal{G}=\sigma (A\times B:A\in \mathcal{F},B\in \mathcal{G}).$$ |

More generally, the product $\sigma $-algebra can be defined for an arbitrary number of measurable spaces $({E}_{i},{\mathcal{F}}_{i})$, where $i$ runs over an index set^{} $I$. The product ${\prod}_{i}{\mathcal{F}}_{i}$ is the $\sigma $-algebra on the generalized cartesian product ${\prod}_{i}{E}_{i}$ generated by sets of the form ${\prod}_{i}{A}_{i}$ where ${A}_{i}\in {\mathcal{F}}_{i}$ for all $i$, and ${A}_{i}={E}_{i}$ for all but finitely many $i$.
If ${\pi}_{j}:{\prod}_{i}{E}_{i}\to {E}_{j}$ are the projection maps, then this is the smallest $\sigma $-algebra with respect to which each ${\pi}_{j}$ is measurable (http://planetmath.org/MeasurableFunctions).

Title | product $\sigma $-algebra |
---|---|

Canonical name | Productsigmaalgebra |

Date of creation | 2013-03-22 18:47:21 |

Last modified on | 2013-03-22 18:47:21 |

Owner | gel (22282) |

Last modified by | gel (22282) |

Numerical id | 6 |

Author | gel (22282) |

Entry type | Definition |

Classification | msc 28A60 |

Synonym | product sigma-algebra |

Related topic | ProductMeasure |

Related topic | InfiniteProductMeasure |