# 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

## 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)

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