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 ${\displaystyle X,}$ the cylindrical σ-algebra ${\displaystyle {\mathfrak {A}}(X,X')}$ is defined to be the coarsest σ-algebra (that is, the one with the fewest measurable sets) such that every continuous linear function on ${\displaystyle X}$ is a measurable function. In general, ${\displaystyle {\mathfrak {A}}(X,X')}$ is not the same as the Borel σ-algebra on ${\displaystyle X,}$ which is the coarsest σ-algebra that contains all open subsets of ${\displaystyle X.}$

Definition

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

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

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

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

is called the cylindrical algebra. Equivalently one can also look at the open cylinder sets and get the same algebra. Then ${\displaystyle {\mathfrak {A}}(N,M)=\sigma (\operatorname {Cyl} (N,M))}$ is the cylindrical σ-algebra.[4]

Properties

• Let ${\displaystyle X}$ a Hausdorff locally convex space which is also a hereditarily Lindelöf space, then

${\displaystyle {\mathfrak {A}}(X,X')={\mathcal {B}}(X).}$[5]

See also

• Cylinder set – natural basic set in product spacesPages displaying wikidata descriptions as a fallback
• Cylinder set measure – way to generate a measure over product spacesPages displaying wikidata descriptions as a fallback

References

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.

• 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)