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A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027620/c0276201.png" /> in a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027620/c0276202.png" /> over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027620/c0276203.png" /> of real numbers given by an equation
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027620/c0276204.png" /></td> </tr></table>
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027620/c0276205.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027620/c0276206.png" /> are linear functions defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027620/c0276207.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027620/c0276208.png" /> is a Borel set in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027620/c0276209.png" />-dimensional space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027620/c02762010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027620/c02762011.png" />.
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A set $  S $
 +
in a vector space  $  L $
 +
over the field  $  \mathbf R $
 +
of real numbers given by an equation
  
The collection of all cylinder sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027620/c02762012.png" /> forms an algebra of sets, the so-called cylinder algebra. The smallest <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027620/c02762013.png" />-algebra of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027620/c02762014.png" /> containing the cylinder sets is called the cylinder <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027620/c02762016.png" />-algebra.
+
$$
 +
S  \equiv \
 +
S _ {\{ A; F _ {1}  \dots F _ {n} \} }  = \
 +
\{ {x \in L } : {( F _ {1} ( x) \dots F _ {n} ( x)) \in A } \}
 +
,
 +
$$
  
When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027620/c02762017.png" /> is a topological vector space, one considers only cylinder sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027620/c02762018.png" /> that are defined by collections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027620/c02762019.png" /> of continuous linear functions. Here by the cylinder algebra and the cylinder <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027620/c02762020.png" />-algebra one understands the corresponding collection of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027620/c02762021.png" /> that are generated by precisely such cylinder sets. In the important special case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027620/c02762022.png" /> is the topological dual of some topological vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027620/c02762023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027620/c02762024.png" />, cylinder sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027620/c02762025.png" /> are defined by means of *-weakly continuous linear functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027620/c02762026.png" />, that is, functions of the form
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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 , . . . $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027620/c02762027.png" /></td> </tr></table>
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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.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027620/c02762028.png" /> is an arbitrary element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027620/c02762029.png" />.
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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====
 
====Comments====
In a somewhat more general context, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027620/c02762030.png" /> be a product of (topological) spaces. An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027620/c02762032.png" />-cylinder set, or simply a cylinder set, in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027620/c02762033.png" /> is a set of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027620/c02762034.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027620/c02762035.png" /> is a finite subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027620/c02762036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027620/c02762037.png" /> is a subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027620/c02762038.png" />.
+
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  $.

Latest revision as of 17:32, 5 June 2020


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 $.

How to Cite This Entry:
Cylinder set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cylinder_set&oldid=46574
This article was adapted from an original article by R.A. Minlos (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article