In mathematics — specifically, in measure theory and functional analysis — the cylindrical σ-algebra or product σ-algebra 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 {\displaystyle X} and its dual space of continuous linear functionals X ′ , {\displaystyle X',} the cylindrical σ-algebra A ( X , X ′ ) {\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 X {\displaystyle X} is a measurable function. In general, A ( X , X ′ ) {\displaystyle {\mathfrak {A}}(X,X')} is not the same as the Borel σ-algebra on X , {\displaystyle X,} which is the coarsest σ-algebra that contains all open subsets of X . {\displaystyle X.}
Definition Consider two topological vector spaces N {\displaystyle N} and M {\displaystyle M} with dual pairing ⟨ , ⟩ := ⟨ , ⟩ N , M {\displaystyle \langle ,\rangle :=\langle ,\rangle _{N,M}} , then we can define the so called Borel cylinder sets
C f 1 , … , f m , B = { x ∈ N : ( ⟨ x , f 1 ⟩ , … , ⟨ x , f m ⟩ ) ∈ B } {\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 f 1 , … , f m ∈ M {\displaystyle f_{1},\dots ,f_{m}\in M} and B ∈ B ( R m ) {\displaystyle B\in {\mathcal {B}}(\mathbb {R} ^{m})} . The family of all these sets is denoted as A f 1 , … , f n {\displaystyle {\mathfrak {A}}_{f_{1},\dots ,f_{n}}} . Then
Cyl ( N , M ) = ⨂ n A f 1 , … , f n {\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. The cylindrical σ-algebra A ( N , M ) = σ ( Cyl ( N , M ) ) {\displaystyle {\mathfrak {A}}(N,M)=\sigma (\operatorname {Cyl} (N,M))} is the σ-algebra generated by the cylindrical algebra.
Properties Let X {\displaystyle X} a Hausdorff locally convex space which is also a hereditarily Lindelöf space, then
A ( X , X ′ ) = B ( X ) . {\displaystyle {\mathfrak {A}}(X,X')={\mathcal {B}}(X).}
See also Cylinder set – Natural basic set in product spaces Cylinder set measure Measure theory in topological vector spaces
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) 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)