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 $X$ and its dual space of continuous linear functionals $X',$ the cylindrical σ-algebra ${\mathfrak {A}}(X,X')$ 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, ${\mathfrak {A}}(X,X')$ is not the same as the Borel σ-algebra on $X,$ which is the coarsest σ-algebra that contains all open subsets of $X.$

Definition

Consider two topological vector spaces $N$ and $M$ with dual pairing $\langle ,\rangle :=\langle ,\rangle _{N,M}$ , then we can define the so called Borel cylinder sets

C_{f_{1},\dots ,f_{m},B}=\{x\in N\colon (\langle x,f_{1}\rangle ,\dots ,\langle x,f_{m}\rangle )\in B\}

for some $f_{1},\dots ,f_{m}\in M$ and $B\in {\mathcal {B}}(\mathbb {R} ^{m})$ . The family of all these sets is denoted as ${\mathfrak {A}}_{f_{1},\dots ,f_{n}}$ . Then

\operatorname {Cyl} (N,M)=\bigotimes _{n}{\mathfrak {A}}_{f_{1},\dots ,f_{n}}

is called the cylindrical algebra. Equivalently one can also look at the open cylinder sets and get the same algebra.

The cylindrical σ-algebra ${\mathfrak {A}}(N,M)=\sigma (\operatorname {Cyl} (N,M))$ is the σ-algebra generated by the cylinderical algebra.^[4]

Properties

Let $X$ a Hausdorff locally convex space which is also a hereditarily Lindelöf space, then

{\mathfrak {A}}(X,X')={\mathcal {B}}(X).

^[5]

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.
^ Richard M. Dudley, Jacob Feldman und Lucien Le Cam (1971), Princeton University (ed.), "On Seminorms and Probabilities, and Abstract Wiener Spaces", Annals of Mathematics, vol. 93, no. 2, pp. 390–392
^ Mitoma, Itaru; Okada, Susumu; Okazaki, Yoshiaki (1977). "Cylindrical σ-algebra and cylindrical measure". Osaka Journal of Mathematics. 14 (3). Osaka University and Osaka Metropolitan University, Departments of Mathematics: 640.

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)

Lunardi, Alessandra; Miranda, Michele; Pallara, Diego (2016), Inﬁnite Dimensional Analysis, Lecture Notes, 19th Internet Seminar, Dipartimento di Matematica e Informatica Università degli Studi di Ferrara (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.

[4] Richard M. Dudley, Jacob Feldman und Lucien Le Cam (1971), Princeton University (ed.), "On Seminorms and Probabilities, and Abstract Wiener Spaces", Annals of Mathematics, vol. 93, no. 2, pp. 390–392

[5] Mitoma, Itaru; Okada, Susumu; Okazaki, Yoshiaki (1977). "Cylindrical σ-algebra and cylindrical measure". Osaka Journal of Mathematics. 14 (3). Osaka University and Osaka Metropolitan University, Departments of Mathematics: 640.

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v t e Banach space topics
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Definition

Properties

See also

References