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 $.

Comments

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 $.