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 $X,$ 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

^ Gine, Evarist; Nickl, Richard (2016). Mathematical Foundations of Infinite-Dimensional Statistical Models. Cambridge University Press. p. 16.
^ Athreya, Krishna; Lahiri, Soumendra (2006). Measure Theory and Probability Theory. Springer. pp. 202–203.
^ 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)

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[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.

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Cylindrical σ-algebra

See also

References

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