# 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 Productsigmaalgebra 2013-03-22 18:47:21 2013-03-22 18:47:21 gel (22282) gel (22282) 6 gel (22282) Definition msc 28A60 product sigma-algebra ProductMeasure InfiniteProductMeasure