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Difference between revisions of "Pre-measure"

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m (fixing spaces)
 
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of the form  $  \mathfrak A = \cup _ {\alpha \in A }  \mathfrak B _  \alpha  $,  
 
of the form  $  \mathfrak A = \cup _ {\alpha \in A }  \mathfrak B _  \alpha  $,  
 
where  $  \mathfrak B _  \alpha  $
 
where  $  \mathfrak B _  \alpha  $
is a family of  $  \sigma $-
+
is a family of  $  \sigma $-algebras of  $  \Omega $,  
algebras of  $  \Omega $,  
 
 
labelled by the elements of some partially ordered set  $  A $,  
 
labelled by the elements of some partially ordered set  $  A $,  
 
such that  $  \mathfrak B _ {\alpha _ {1}  } \subset  \mathfrak B _ {\alpha _ {2}  } $
 
such that  $  \mathfrak B _ {\alpha _ {1}  } \subset  \mathfrak B _ {\alpha _ {2}  } $
 
if  $  \alpha _ {1} < \alpha _ {2} $,  
 
if  $  \alpha _ {1} < \alpha _ {2} $,  
while the restriction of the measure to any  $  \sigma $-
+
while the restriction of the measure to any  $  \sigma $-algebra  $  \mathfrak B _  \alpha  $
algebra  $  \mathfrak B _  \alpha  $
 
 
is countably additive. E.g., if  $  \Omega $
 
is countably additive. E.g., if  $  \Omega $
 
is a Hausdorff space,  $  A $
 
is a Hausdorff space,  $  A $
 
is the family of all compacta, ordered by inclusion,  $  \mathfrak B _  \alpha  $,  
 
is the family of all compacta, ordered by inclusion,  $  \mathfrak B _  \alpha  $,  
 
$  \alpha \in A $,  
 
$  \alpha \in A $,  
is the  $  \sigma $-
+
is the  $  \sigma $-algebra of all Borel subsets of the compactum  $  \alpha $
algebra of all Borel subsets of the compactum  $  \alpha $
 
 
and  $  C _ {0} ( \Omega ) $
 
and  $  C _ {0} ( \Omega ) $
 
is the space of all continuous functions on  $  \Omega $
 
is the space of all continuous functions on  $  \Omega $
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ordered by inclusion, and let  $  \mathfrak B _  \alpha  $,  
 
ordered by inclusion, and let  $  \mathfrak B _  \alpha  $,  
 
$  \alpha \in A $,  
 
$  \alpha \in A $,  
be the least  $  \sigma $-
+
be the least  $  \sigma $-algebra relative to which all linear functionals  $  \phi \in \alpha $
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 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 $
 
are called cylindrical sets, and any pre-measure on  $  \mathfrak A $
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for all  $  A \in {\mathcal R} $;
 
for all  $  A \in {\mathcal R} $;
  
ii)  $  \mu ( \cup _ {n=} 1 ^  \infty  A _ {n} ) = \sum _ {n=} 1 ^  \infty  \mu ( A _ {n} ) $
+
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} $
 
for every countable sequence of pairwise disjoint subsets  $  A _ {n} \in {\mathcal R} $
 
such that  $  \cup A _ {n} \in {\mathcal R} $.
 
such that  $  \cup A _ {n} \in {\mathcal R} $.

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=48276
This article was adapted from an original article by R.A. Minlos (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article