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 either product measure or probability measure of random variables on Banach spaces.

For a product space, the cylinder σ-algebra is the one which is generated by cylinder sets. As for products of countable length, the cylindrical σ-algebra is the product σ-algebra.[1]

In the context of 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


  • 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)
  1. Gerald B Folland (2013). Real Analysis: Modern Techniques and Their Applications. John Wiley & Sons. p. 23. ISBN 0471317160.

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