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 $X',$ the cylindrical σ-algebra $\mathfrak(X,X')$ 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, $\mathfrak(X,X')$ 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 $\langle,\rangle:=\langle,\rangle_$, then we can define the so called Borel cylinder sets
:$C_=\$
for some $f_1,\dots,f_m\in M$ and $B\in \mathcal(\mathbb^m)$. The family of all these sets is denoted as $\mathfrak_$.
Then
:$\operatorname(N,M)=\bigotimes_n \mathfrak_$
is called the cylindrical algebra. Equivalently one can also look at the open cylinder sets and get the same algebra.

The cylindrical σ-algebra $\mathfrak(N,M)=\sigma(\operatorname(N,M))$ 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
::$\mathfrak(X,X')=\mathcal(X).$

See also

*
*

References

* (See chapter 2)

* (See chapter 2)

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Banach spaces
Functional analysis
Measure theory