Cylinder set
A set in a vector space
over the field
of real numbers given by an equation
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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
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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
.
Cylinder set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cylinder_set&oldid=46574