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A finitely-additive [[Measure|measure]] with real or complex values on some space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074300/p0743001.png" /> having the property that it is defined on an algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074300/p0743002.png" /> of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074300/p0743003.png" /> of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074300/p0743004.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074300/p0743005.png" /> is a family of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074300/p0743006.png" />-algebras of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074300/p0743007.png" />, labelled by the elements of some partially ordered set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074300/p0743008.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074300/p0743009.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074300/p07430010.png" />, while the restriction of the measure to any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074300/p07430011.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074300/p07430012.png" /> is countably additive. E.g., if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074300/p07430013.png" /> is a Hausdorff space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074300/p07430014.png" /> is the family of all compacta, ordered by inclusion, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074300/p07430015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074300/p07430016.png" />, is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074300/p07430017.png" />-algebra of all Borel subsets of the compactum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074300/p07430018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074300/p07430019.png" /> is the space of all continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074300/p07430020.png" /> with compact support, then every linear functional on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074300/p07430021.png" /> that is continuous in the topology of uniform convergence in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074300/p07430022.png" /> generates a pre-measure on the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074300/p07430023.png" />.
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074300/p07430024.png" /> be a locally convex linear space, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074300/p07430025.png" /> be the set of finite-dimensional subspaces of the dual space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074300/p07430026.png" />, ordered by inclusion, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074300/p07430027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074300/p07430028.png" />, be the least <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074300/p07430029.png" />-algebra relative to which all linear functionals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074300/p07430030.png" /> are measurable. The sets of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074300/p07430031.png" /> are called cylindrical sets, and any pre-measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074300/p07430032.png" /> is called a [[Cylindrical measure|cylindrical measure]] (or quasi-measure). A positive-definite functional on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074300/p07430033.png" /> that is continuous on any finite-dimensional subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074300/p07430034.png" /> is the characteristic function (Fourier transform) of a finite non-negative pre-measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074300/p07430035.png" />.
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A finitely-additive [[Measure|measure]] with real or complex values on some space  $  \Omega $
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having the property that it is defined on an algebra  $  \mathfrak A $
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of subsets of  $  \Omega $
 +
of the form  $  \mathfrak A = \cup _ {\alpha \in A }  \mathfrak B _  \alpha  $,
 +
where  $  \mathfrak B _  \alpha  $
 +
is a family of  $  \sigma $-algebras of  $  \Omega $,
 +
labelled by the elements of some partially ordered set  $  A $,
 +
such that  $  \mathfrak B _ {\alpha _ {1}  } \subset  \mathfrak B _ {\alpha _ {2}  } $
 +
if  $  \alpha _ {1} < \alpha _ {2} $,
 +
while the restriction of the measure to any  $  \sigma $-algebra  $  \mathfrak B _  \alpha  $
 +
is countably additive. E.g., if  $  \Omega $
 +
is a Hausdorff space,  $  A $
 +
is the family of all compacta, ordered by inclusion,  $  \mathfrak B _  \alpha  $,
 +
$  \alpha \in A $,
 +
is the  $  \sigma $-algebra of all Borel subsets of the compactum  $  \alpha $
 +
and  $  C _ {0} ( \Omega ) $
 +
is the space of all continuous functions on  $  \Omega $
 +
with compact support, then every linear functional on  $  C _ {0} ( \Omega ) $
 +
that is continuous in the topology of uniform convergence in  $  C _ {0} ( \Omega ) $
 +
generates a pre-measure on the algebra  $  \mathfrak A = \cup _ {\alpha \in A }  \mathfrak B _  \alpha  $.
 +
 
 +
Let  $  \Omega $
 +
be a locally convex linear space, let $  A $
 +
be the set of finite-dimensional subspaces of the dual space $  \Omega  ^  \prime  $,  
 +
ordered by inclusion, and let $  \mathfrak B _  \alpha  $,  
 +
$  \alpha \in A $,  
 +
be the least $  \sigma $-algebra relative to which all linear functionals $  \phi \in \alpha $
 +
are measurable. The sets of the algebra $  \mathfrak A = \cup _ {\alpha \in A }  \mathfrak B $
 +
are called cylindrical sets, and any pre-measure on $  \mathfrak A $
 +
is called a [[Cylindrical measure|cylindrical measure]] (or quasi-measure). A positive-definite functional on $  \Omega  ^  \prime  $
 +
that is continuous on any finite-dimensional subspace $  \alpha \subset  \Omega $
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is the characteristic function (Fourier transform) of a finite non-negative pre-measure on $  \Omega $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Integration" , Addison-Wesley  (1975)  pp. Chapt.6;7;8  (Translated from French)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Integration" , Addison-Wesley  (1975)  pp. Chapt.6;7;8  (Translated from French)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The term  "pre-measure"  is also used in the following, related but somewhat different, sense. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074300/p07430036.png" /> be a ring of sets on some space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074300/p07430037.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074300/p07430038.png" /> a numerical function defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074300/p07430039.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074300/p07430040.png" /> is a pre-measure if
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The term  "pre-measure"  is also used in the following, related but somewhat different, sense. Let $  {\mathcal R} $
 +
be a ring of sets on some space $  \Omega $,  
 +
and $  \mu $
 +
a numerical function defined on $  {\mathcal R} $.  
 +
Then $  \mu $
 +
is a pre-measure if
  
i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074300/p07430041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074300/p07430042.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074300/p07430043.png" />;
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i) $  \mu ( \emptyset ) = 0 $,  
 +
$  \mu ( A) \geq  0 $
 +
for all $  A \in {\mathcal R} $;
  
ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074300/p07430044.png" /> for every countable sequence of pairwise disjoint subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074300/p07430045.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074300/p07430046.png" />.
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ii) $  \mu ( \cup _ {n= 1}  ^  \infty  A _ {n} ) = \sum _ {n= 1}  ^  \infty  \mu ( A _ {n} ) $
 +
for every countable sequence of pairwise disjoint subsets $  A _ {n} \in {\mathcal R} $
 +
such that $  \cup A _ {n} \in {\mathcal R} $.
  
If ii) only holds for finite disjoint sequences, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074300/p07430047.png" /> is called a content. Not every content is a pre-measure.
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If ii) only holds for finite disjoint sequences, $  \mu $
 +
is called a content. Not every content is a pre-measure.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Bauer,  "Probability theory and elements of measure theory" , Holt, Rinehart &amp; Winston  (1972)  pp. 13ff  (Translated from German)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Bauer,  "Probability theory and elements of measure theory" , Holt, Rinehart &amp; Winston  (1972)  pp. 13ff  (Translated from German)</TD></TR></table>

Latest revision as of 04:40, 9 May 2022


A finitely-additive measure with real or complex values on some space $ \Omega $ having the property that it is defined on an algebra $ \mathfrak A $ of subsets of $ \Omega $ of the form $ \mathfrak A = \cup _ {\alpha \in A } \mathfrak B _ \alpha $, where $ \mathfrak B _ \alpha $ is a family of $ \sigma $-algebras of $ \Omega $, labelled by the elements of some partially ordered set $ A $, such that $ \mathfrak B _ {\alpha _ {1} } \subset \mathfrak B _ {\alpha _ {2} } $ if $ \alpha _ {1} < \alpha _ {2} $, while the restriction of the measure to any $ \sigma $-algebra $ \mathfrak B _ \alpha $ is countably additive. E.g., if $ \Omega $ is a Hausdorff space, $ A $ is the family of all compacta, ordered by inclusion, $ \mathfrak B _ \alpha $, $ \alpha \in A $, is the $ \sigma $-algebra of all Borel subsets of the compactum $ \alpha $ and $ C _ {0} ( \Omega ) $ is the space of all continuous functions on $ \Omega $ with compact support, then every linear functional on $ C _ {0} ( \Omega ) $ that is continuous in the topology of uniform convergence in $ C _ {0} ( \Omega ) $ generates a pre-measure on the algebra $ \mathfrak A = \cup _ {\alpha \in A } \mathfrak B _ \alpha $.

Let $ \Omega $ be a locally convex linear space, let $ A $ be the set of finite-dimensional subspaces of the dual space $ \Omega ^ \prime $, ordered by inclusion, and let $ \mathfrak B _ \alpha $, $ \alpha \in A $, be the least $ \sigma $-algebra relative to which all linear functionals $ \phi \in \alpha $ are measurable. The sets of the algebra $ \mathfrak A = \cup _ {\alpha \in A } \mathfrak B $ are called cylindrical sets, and any pre-measure on $ \mathfrak A $ is called a cylindrical measure (or quasi-measure). A positive-definite functional on $ \Omega ^ \prime $ that is continuous on any finite-dimensional subspace $ \alpha \subset \Omega $ is the characteristic function (Fourier transform) of a finite non-negative pre-measure on $ \Omega $.

References

[1] N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated from French)

Comments

The term "pre-measure" is also used in the following, related but somewhat different, sense. Let $ {\mathcal R} $ be a ring of sets on some space $ \Omega $, and $ \mu $ a numerical function defined on $ {\mathcal R} $. Then $ \mu $ is a pre-measure if

i) $ \mu ( \emptyset ) = 0 $, $ \mu ( A) \geq 0 $ for all $ A \in {\mathcal R} $;

ii) $ \mu ( \cup _ {n= 1} ^ \infty A _ {n} ) = \sum _ {n= 1} ^ \infty \mu ( A _ {n} ) $ for every countable sequence of pairwise disjoint subsets $ A _ {n} \in {\mathcal R} $ such that $ \cup A _ {n} \in {\mathcal R} $.

If ii) only holds for finite disjoint sequences, $ \mu $ is called a content. Not every content is a pre-measure.

References

[a1] H. Bauer, "Probability theory and elements of measure theory" , Holt, Rinehart & Winston (1972) pp. 13ff (Translated from German)
How to Cite This Entry:
Pre-measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pre-measure&oldid=18650
This article was adapted from an original article by R.A. Minlos (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article