Cylindrical σ-algebra

In mathematics &mdash; specifically, in measure theory and functional analysis &mdash; the cylindrical σ-algebra or product σ-algebra is a type of σ-algebra which is often used when studying product measures or probability measures of random variables on Banach spaces.

For a product space, the cylinder σ-algebra is the one that is generated by cylinder sets.

In the context of a Banach space $$X,$$ the cylindrical σ-algebra $$\operatorname{Cyl}(X)$$ is defined to be the coarsest σ-algebra (that is, the one with the fewest measurable sets) such that every continuous linear function on $$X$$ is a measurable function. In general, $$\operatorname{Cyl}(X)$$ is not the same as the Borel σ-algebra on $$X,$$ which is the coarsest σ-algebra that contains all open subsets of $$X.$$