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 $B\in\mathcal{G}$ \begin{equation*} \mathcal{F}\times\mathcal{G}=\sigma\left(A\times B\colon A\in\mathcal{F},B\in\mathcal{G}\right). \end{equation*}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 $\prod_iE_i$ generated by sets of the form $\prod_iA_i$ where $A_i\in\mathcal{F}_i$ for all $i$ and $A_i=E_i$ for all but finitely many $i$ If $\pi_j\colon\prod_iE_i\rightarrow E_j$ are the projection maps, then this is the smallest $\sigma$ algebra with respect to which each $\pi_j$ is measurable.