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 [math]\displaystyle{ X, }[/math] the cylindrical σ-algebra [math]\displaystyle{ \operatorname{Cyl}(X) }[/math] is defined to be the coarsest σ-algebra (that is, the one with the fewest measurable sets) such that every continuous linear function on [math]\displaystyle{ X }[/math] is a measurable function. In general, [math]\displaystyle{ \operatorname{Cyl}(X) }[/math] is not the same as the Borel σ-algebra on [math]\displaystyle{ X, }[/math] which is the coarsest σ-algebra that contains all open subsets of [math]\displaystyle{ X. }[/math]
See also
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 (1991). Probability in Banach spaces. Berlin: Springer-Verlag. pp. xii+480. ISBN 3-540-52013-9. (See chapter 2)
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