Cylindrical σ-algebra

Cylindrical σ-algebra

In mathematics — specifically, in measure theory and functional analysis — the cylindrical σ-algebra is a σ-algebra often used in the study of probability measures and random variables on Banach spaces. For 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.

References

  • Ledoux, Michel; Talagrand, Michel (1991). Probability in Banach spaces. Berlin: Springer-Verlag. pp. xii+480. ISBN 3-540-52013-9. MR1102015.  (See chapter 2)

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