Cylindrical σ-algebra explained

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

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.

References

• Book: Ledoux. Michel. Talagrand . Michel . Michel Talagrand. Probability in Banach spaces. Springer-Verlag. Berlin. 1991. xii+480. 3-540-52013-9. 1102015. (See chapter 2)

Notes and References

1. Book: Gine . Evarist . Nickl . Richard . Mathematical Foundations of Infinite-Dimensional Statistical Models . 2016 . Cambridge University Press . 16.
2. Book: Athreya . Krishna . Lahiri . Soumendra . Measure Theory and Probability Theory . 2006 . Springer . 202–203.
3. Book: Cohn . Donald . Measure Theory . 2013 . Birkhauser . 365 . Second.