# Cylinder set

A set $S$ in a vector space $L$ over the field $\mathbf R$ of real numbers given by an equation

$$S \equiv \ S _ {\{ A; F _ {1} \dots F _ {n} \} } = \ \{ {x \in L } : {( F _ {1} ( x) \dots F _ {n} ( x)) \in A } \} ,$$

where $F \in L ^ {*}$, $i = 1, 2 \dots$ are linear functions defined on $L$ and $A \subset \mathbf R ^ {n}$ is a Borel set in the $n$- dimensional space $\mathbf R ^ {n}$, $n = 1, 2 , . . .$.

The collection of all cylinder sets in $L$ forms an algebra of sets, the so-called cylinder algebra. The smallest $\sigma$- algebra of subsets of $L$ containing the cylinder sets is called the cylinder $\sigma$- algebra.

When $L$ is a topological vector space, one considers only cylinder sets $S _ {\{ A; F _ {1} \dots F _ {n} \} }$ that are defined by collections $\{ F _ {1} \dots F _ {n} \}$ of continuous linear functions. Here by the cylinder algebra and the cylinder $\sigma$- algebra one understands the corresponding collection of subsets of $L$ that are generated by precisely such cylinder sets. In the important special case when $L$ is the topological dual of some topological vector space $M$, $L = M ^ { \prime }$, cylinder sets in $L$ are defined by means of *-weakly continuous linear functions on $L$, that is, functions of the form

$$F _ \phi ( x) = x ( \phi ),\ \ x \in L,$$

where $\phi$ is an arbitrary element of $M$.

In a somewhat more general context, let $X = \prod _ {\alpha \in A } X _ \alpha$ be a product of (topological) spaces. An $n$- cylinder set, or simply a cylinder set, in $X$ is a set of the form $U \times \prod _ {\alpha \notin S } X _ \alpha$ where $S$ is a finite subset of $A$ and $U$ is a subset of $\prod _ {\alpha \in S } X _ \alpha$.