In mathematics — specifically, in measure theory and functional analysis — the cylindrical σ-algebra[1] or product σ-algebra[2][3] 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
the cylindrical σ-algebra
is defined to be the coarsest σ-algebra (that is, the one with the fewest measurable sets) such that every continuous linear function on
is a measurable function.
is not the same as the Borel σ-algebra on
which is the coarsest σ-algebra that contains all open subsets of
Consider two topological vector spaces
, then we can define the so called Borel cylinder sets for some
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.
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