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 an [arbitrary">Σ-algebra">generated by an [arbitrary family|generated] by cylinder sets.
In the context of a Banach space and its dual space of continuous [linear functional]s the cylindrical σ-algebra is defined to be the coarsest σ-algebra such that every continuous linear function on is a measurable function. In general, is not the same as the Borel σ-algebra on which is the coarsest σ-algebra that contains all open subsets of

Definition

Consider two topological [vector space]s and with dual pairing, then we can define the so called Borel cylinder sets
for some and. The family of all these sets is denoted as.
Then
is called the cylindrical algebra. Equivalently one can also look at the open cylinder sets and get the same algebra.
The cylindrical σ-algebra is the σ-algebra generated by the cylindrical algebra.

Properties