product $\sigma$-algebra

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

$$ℱ\times 𝒢=\sigma (A\times B:A\in ℱ,B\in 𝒢).$$

More generally, the product $\sigma$-algebra can be defined for an arbitrary number of measurable spaces $(E_i,{ℱ}_i)$, where $i$ runs over an index set $I$. The product ${\prod }_i{ℱ}_i$ is the $\sigma$-algebra on the generalized cartesian product ${\prod }_i{E}_i$ generated by sets of the form ${\prod }_i{A}_i$ where $A_i\in {ℱ}_i$ 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).