Cylindrical σ-algebra

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

See also

• Cylinder set – natural basic set in product spacesPages displaying wikidata descriptions as a fallback
• Cylinder set measure – way to generate a measure over product spacesPages displaying wikidata descriptions as a fallback

References

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

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