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 ${\displaystyle X}$ and its dual space of continuous linear functionals ${\displaystyle X',}$ 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.

The cylindrical σ-algebra ${\displaystyle {\mathfrak {A}}(N,M)=\sigma (\operatorname {Cyl} (N,M))}$ is the σ-algebra generated by the cylinderical 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
• Cylinder set measure
• Measure theory in topological vector spaces