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 Cyl(X) is defined to be the coarsest σ-algebra (i.e. the one with the fewest measurable sets) such that every continuous linear function on X is a measurable function. In general, 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.
See also
References
- 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)
- ^ 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.
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