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 $\operatorname {Cyl} (X)$ is defined to be the coarsest σ-algebra (that is, the one with the fewest measurable sets) such that every continuous linear function on $X$ is a measurable function. In general, $\operatorname {Cyl} (X)$ 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

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; Talagrand, Michel (1991). Probability in Banach spaces. Berlin: Springer-Verlag. pp. xii+480. ISBN 3-540-52013-9. MR 1102015. (See chapter 2)

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[1] Gine, Evarist; Nickl, Richard (2016). Mathematical Foundations of Infinite-Dimensional Statistical Models. Cambridge University Press. p. 16.

[2] Athreya, Krishna; Lahiri, Soumendra (2006). Measure Theory and Probability Theory. Springer. pp. 202–203.

[3] Cohn, Donald (2013). Measure Theory (Second ed.). Birkhauser. p. 365.

Banach space topics
Types of Banach spaces	Asplund Banach list Banach lattice Grothendieck Hilbert Inner product space Polarization identity (Polynomially) Reflexive Riesz L-semi-inner product (B Strictly Uniformly) convex Uniformly smooth (Injective Projective) Tensor product (of Hilbert spaces)
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Operator theory	Banach algebras C-algebras Operator space Spectrum C-algebra radius Spectral theory of ODEs Spectral theorem Polar decomposition Singular value decomposition
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Analysis	Abstract Wiener space Banach manifold bundle Bochner space Convex series Differentiation in Fréchet spaces Derivatives Fréchet Gateaux functional holomorphic quasi Integrals Bochner Dunford Gelfand–Pettis regulated Paley–Wiener weak Functional calculus Borel continuous holomorphic Measures Lebesgue Projection-valued Vector Weakly / Strongly measurable function
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Applications	Differential operator Finite element method Mathematical formulation of quantum mechanics Ordinary Differential Equations (ODEs) Validated numerics

Cylindrical σ-algebra

See also

References