Cylindrical σ-algebra

In mathematics — specifically, in measure theory and functional analysis — 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 ${\displaystyle X}$ and its dual space of continuous linear functionals ${\displaystyle X',}$ the cylindrical σ-algebra ${\displaystyle {\mathfrak {A}}(X,X')}$ is defined to be the coarsest σ-algebra (that is, the one with the fewest measurable sets) such that every continuous linear function on ${\displaystyle X}$ is a measurable function. In general, ${\displaystyle {\mathfrak {A}}(X,X')}$ is not the same as the Borel σ-algebra on ${\displaystyle X,}$ which is the coarsest σ-algebra that contains all open subsets of ${\displaystyle X.}$