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A cylindrical measure in measure theory on topological vector spaces is a finitely-additive measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027640/c0276401.png" /> defined on the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027640/c0276402.png" /> of cylinder sets in a topological vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027640/c0276403.png" />, that is, sets of the form
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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/c027640/c0276404.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027640/c0276405.png" /> — the Borel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027640/c0276406.png" />-algebra of subsets of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027640/c0276407.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027640/c0276408.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027640/c0276409.png" /> are linear functionals on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027640/c02764010.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027640/c02764011.png" /> is the mapping
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A cylindrical measure in measure theory on topological vector spaces is a finitely-additive measure  $  \mu $
 +
defined on the algebra $  \mathfrak A ( E) $
 +
of cylinder sets in a topological vector space $  E $,  
 +
that is, sets of the form
  
<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/c027640/c02764012.png" /></td> </tr></table>
+
$$ \tag{* }
 +
= F _ {\phi _ {1}  \dots \phi _ {n} } ^ { - 1 } ( B),
 +
$$
  
Here it is assumed that the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027640/c02764013.png" /> to any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027640/c02764014.png" />-subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027640/c02764015.png" /> of sets of the form (*) with a fixed collection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027640/c02764016.png" /> of functionals is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027640/c02764017.png" />-additive measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027640/c02764018.png" /> (other names are pre-measure, quasi-measure).
+
where  $  B \in \mathfrak B ( \mathbf R  ^ {n} ) $—
 +
the Borel  $  \sigma $-
 +
algebra of subsets of the space  $  \mathbf R  ^ {n} $,
 +
$  n = 1, 2 ,\dots $;
 +
$  \phi _ {1} \dots \phi _ {n} $
 +
are linear functionals on $  E $,  
 +
and  $  F _ {\phi _ {1}  \dots \phi _ {n} } $
 +
is the mapping
  
In the theory of functions of several real variables a cylindrical measure is a special case of the [[Hausdorff measure|Hausdorff measure]], defined on the Borel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027640/c02764019.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027640/c02764020.png" /> of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027640/c02764021.png" /> by means of the formula
+
$$
 +
E  \rightarrow  \mathbf R  ^ {n} : \
 +
x  \rightarrow  \{ \phi _ {1} ( x) \dots
 +
\phi _ {n} ( x) \}  \in  \mathbf R  ^ {n} ,\ \
 +
x \in E.
 +
$$
  
<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/c027640/c02764022.png" /></td> </tr></table>
+
Here it is assumed that the restriction of  $  \mu $
 +
to any  $  \sigma $-
 +
subalgebra  $  \mathfrak B _ {\phi _ {1}  \dots \phi _ {n} } ( E) \subset  \mathfrak A ( E) $
 +
of sets of the form (*) with a fixed collection  $  ( \phi _ {1} \dots \phi _ {n} ) $
 +
of functionals is a  $  \sigma $-
 +
additive measure on  $  \mathfrak B _ {\phi _ {1}  \dots \phi _ {n} } $(
 +
other names are pre-measure, quasi-measure).
  
where the lower bound is taken over all finite or countable coverings of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027640/c02764023.png" /> by cylinders <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027640/c02764024.png" /> with spherical bases and axes parallel to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027640/c02764025.png" />-st coordinate axis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027640/c02764026.png" />; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027640/c02764027.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027640/c02764028.png" />-dimensional volume of an axial section of the cylinder <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027640/c02764029.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027640/c02764030.png" /> is the graph of a continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027640/c02764031.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027640/c02764032.png" /> variables defined in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027640/c02764033.png" />:
+
In the theory of functions of several real variables a cylindrical measure is a special case of the [[Hausdorff measure|Hausdorff measure]], defined on the Borel  $  \sigma $-
 +
algebra  $  \mathfrak B ( \mathbf R ^ {n + 1 } ) $
 +
of the space  $  \mathbf R ^ {n + 1 } $
 +
by means of the formula
  
<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/c027640/c02764034.png" /></td> </tr></table>
+
$$
 +
\lambda ( B)  = \
 +
\lim\limits _ {\epsilon \rightarrow 0 } \
 +
\inf _ {\begin{array}{c}
 +
\{ A \} , \\
 +
  \mathop{\rm diam}  A < \epsilon
 +
\end{array}
 +
} \
 +
\left \{ \sum l ( A) \right \} ,
 +
$$
  
then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027640/c02764035.png" /> is the same as the so-called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027640/c02764037.png" />-dimensional variation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027640/c02764038.png" />.
+
where the lower bound is taken over all finite or countable coverings of a set  $  B \in \mathfrak B ( \mathbf R  ^ {n+} 1 ) $
 +
by cylinders  $  A $
 +
with spherical bases and axes parallel to the  $  ( n + 1) $-
 +
st coordinate axis in  $  \mathbf R ^ {n + 1 } $;
 +
here  $  l ( A) $
 +
is the  $  n $-
 +
dimensional volume of an axial section of the cylinder  $  A $.
 +
When  $  B $
 +
is the graph of a continuous function  $  f $
 +
of  $  n $
 +
variables defined in a domain  $  G \subset  \mathbf R  ^ {n} $:
 +
 
 +
$$
 +
= \
 +
\{ {( x _ {1} \dots x _ {n+} 1 ) } : {x _ {n+} 1 =
 +
f ( x _ {1} \dots x _ {n} ) } \}
 +
,
 +
$$
 +
 
 +
then  $  \lambda ( B) $
 +
is the same as the so-called $  n $-
 +
dimensional variation of $  f $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.M. Gel'fand,  N.Ya. Vilenkin,  "Generalized functions. Applications of harmonic analysis" , '''4''' , Acad. Press  (1968)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.G. Vitushkin,  "On higher-dimensional variations" , Moscow  (1955)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.M. Gel'fand,  N.Ya. Vilenkin,  "Generalized functions. Applications of harmonic analysis" , '''4''' , Acad. Press  (1968)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.G. Vitushkin,  "On higher-dimensional variations" , Moscow  (1955)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Concerning the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027640/c02764039.png" />-dimensional variation of a function see [[Variation of a function|Variation of a function]].
+
Concerning the $  n $-
 +
dimensional variation of a function see [[Variation of a function|Variation of a function]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Schwartz,  "Radon measures on arbitrary topological spaces and cylindrical measures" , Oxford Univ. Press  (1973)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Schwartz,  "Radon measures on arbitrary topological spaces and cylindrical measures" , Oxford Univ. Press  (1973)</TD></TR></table>

Latest revision as of 17:32, 5 June 2020


A cylindrical measure in measure theory on topological vector spaces is a finitely-additive measure $ \mu $ defined on the algebra $ \mathfrak A ( E) $ of cylinder sets in a topological vector space $ E $, that is, sets of the form

$$ \tag{* } A = F _ {\phi _ {1} \dots \phi _ {n} } ^ { - 1 } ( B), $$

where $ B \in \mathfrak B ( \mathbf R ^ {n} ) $— the Borel $ \sigma $- algebra of subsets of the space $ \mathbf R ^ {n} $, $ n = 1, 2 ,\dots $; $ \phi _ {1} \dots \phi _ {n} $ are linear functionals on $ E $, and $ F _ {\phi _ {1} \dots \phi _ {n} } $ is the mapping

$$ E \rightarrow \mathbf R ^ {n} : \ x \rightarrow \{ \phi _ {1} ( x) \dots \phi _ {n} ( x) \} \in \mathbf R ^ {n} ,\ \ x \in E. $$

Here it is assumed that the restriction of $ \mu $ to any $ \sigma $- subalgebra $ \mathfrak B _ {\phi _ {1} \dots \phi _ {n} } ( E) \subset \mathfrak A ( E) $ of sets of the form (*) with a fixed collection $ ( \phi _ {1} \dots \phi _ {n} ) $ of functionals is a $ \sigma $- additive measure on $ \mathfrak B _ {\phi _ {1} \dots \phi _ {n} } $( other names are pre-measure, quasi-measure).

In the theory of functions of several real variables a cylindrical measure is a special case of the Hausdorff measure, defined on the Borel $ \sigma $- algebra $ \mathfrak B ( \mathbf R ^ {n + 1 } ) $ of the space $ \mathbf R ^ {n + 1 } $ by means of the formula

$$ \lambda ( B) = \ \lim\limits _ {\epsilon \rightarrow 0 } \ \inf _ {\begin{array}{c} \{ A \} , \\ \mathop{\rm diam} A < \epsilon \end{array} } \ \left \{ \sum l ( A) \right \} , $$

where the lower bound is taken over all finite or countable coverings of a set $ B \in \mathfrak B ( \mathbf R ^ {n+} 1 ) $ by cylinders $ A $ with spherical bases and axes parallel to the $ ( n + 1) $- st coordinate axis in $ \mathbf R ^ {n + 1 } $; here $ l ( A) $ is the $ n $- dimensional volume of an axial section of the cylinder $ A $. When $ B $ is the graph of a continuous function $ f $ of $ n $ variables defined in a domain $ G \subset \mathbf R ^ {n} $:

$$ B = \ \{ {( x _ {1} \dots x _ {n+} 1 ) } : {x _ {n+} 1 = f ( x _ {1} \dots x _ {n} ) } \} , $$

then $ \lambda ( B) $ is the same as the so-called $ n $- dimensional variation of $ f $.

References

[1] I.M. Gel'fand, N.Ya. Vilenkin, "Generalized functions. Applications of harmonic analysis" , 4 , Acad. Press (1968) (Translated from Russian)
[2] A.G. Vitushkin, "On higher-dimensional variations" , Moscow (1955) (In Russian)

Comments

Concerning the $ n $- dimensional variation of a function see Variation of a function.

References

[a1] L. Schwartz, "Radon measures on arbitrary topological spaces and cylindrical measures" , Oxford Univ. Press (1973)
How to Cite This Entry:
Cylindrical measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cylindrical_measure&oldid=46575
This article was adapted from an original article by R.A. Minlos (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article