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\left(A\times B\colon A\in\mathcal{F},B\in% \mathcal{G}\right).

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 $\pi_{j}\colon\prod_{i}E_{i}\rightarrow E_{j}$ 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

product σ-algebra

product $\sigma$ -algebra