Cylindrical σ-algebra

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 $X$ and its dual space of continuous linear functionals $X^{\prime },$ the cylindrical σ-algebra $\mathfrak{𝔄}(X,X^{\prime })$ 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, $\mathfrak{𝔄}(X,X^{\prime })$ 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 $⟨,⟩:=⟨,{⟩}_{N,M}$, then we can define the so called Borel cylinder sets

$C_{f_1,\dots ,f_m,B}=\{x\in N:(⟨x,f_1⟩,\dots ,⟨x,f_m⟩)\in B\}$

for some $f_1,\dots ,f_m\in M$ and $B\in {ℬ}({\mathbb{ℝ}}^m)$. The family of all these sets is denoted as ${\mathfrak{𝔄}}_{f_1,\dots ,f_n}$. Then

$\mathrm{Cyl}(N,M)=\underset{n}{⨂}{\mathfrak{𝔄}}_{f_1,\dots ,f_n}$

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

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

Properties

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

$\mathfrak{𝔄}(X,X^{\prime })={ℬ}(X).$

See also

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References

• (See chapter 2)

• (See chapter 2)

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