Cylinder set

A set in a vector space over the field of real numbers given by an equation

where , are linear functions defined on and is a Borel set in the -dimensional space , .

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

When is a topological vector space, one considers only cylinder sets that are defined by collections of continuous linear functions. Here by the cylinder algebra and the cylinder -algebra one understands the corresponding collection of subsets of that are generated by precisely such cylinder sets. In the important special case when is the topological dual of some topological vector space , , cylinder sets in are defined by means of *-weakly continuous linear functions on , that is, functions of the form

where is an arbitrary element of .

Comments

In a somewhat more general context, let be a product of (topological) spaces. An -cylinder set, or simply a cylinder set, in is a set of the form where is a finite subset of and is a subset of .