Coordinate maps, finite-coordinate cylinders, and the cylinder $\sigma$-algebra

For measurable spaces $(E_i,{\mathcal{E}}_i{)}_{i\in I}$, put $E={\prod }_{i\in I}{E}_i$. For finite $F\subseteq I$, let ${\pi }_F:E\to {\prod }_{i\in F}{E}_i$ be restriction to $F$ (the singleton empty product when $F=\mathrm{\varnothing }$). On ${\prod }_{i\in F}{E}_i$, write $q_i$ for the $i$th coordinate map and define

${⨂}_{i\in F}{\mathcal{E}}_i:=\sigma \{q_i^{-1}(A):i\in F,A\in {\mathcal{E}}_i\}\mathrm{.}$

For $F=\mathrm{\varnothing }$ this is the two-set sigma-algebra on the singleton. If $e:[n]\to F$ is any enumeration, coordinate relabelling identifies this sigma-algebra with the recursively defined finite product sigma-algebra of The product sigma-algebra and its finite iterates: both are generated by the same coordinate inverse images. Thus the notation is independent of the enumeration.

A finite-coordinate cylinder is ${\pi }_F^{-1}(A)$ for $A\in {⨂}_{i\in F}{\mathcal{E}}_i$. The cylinder sigma-algebra is

${\mathcal{C}}_I:=\sigma \{{\pi }_F^{-1}(A):F\subseteq I\text{ finite},A\in {⨂}_{i\in F}{\mathcal{E}}_i\}\mathrm{.}$

This is the product sigma-algebra on the arbitrary product; the empty-support cylinder is either $\mathrm{\varnothing }$ or $E$.