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The Riesz Representation Theorem and Extension of Vector Valued Additive Measures

✍ Scribed by Benedetto Bongiorno; Nicolae Dinculeanu

Publisher
Elsevier Science
Year
2001
Tongue
English
Weight
191 KB
Volume
261
Category
Article
ISSN
0022-247X

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✦ Synopsis

let m β†’ E be a finitely additive measure with finite semivariation, defined on a Ξ΄-ring of subsets of a given set S. A theory of integration of vector-valued functions f S β†’ E, applicable to the stochastic integration in Banach spaces, is developed in [6, Sect. 5].

Many times a measure m is defined on a ring (rather than on a Ξ΄ring). In order to apply the above integration theory, we have to extend the measure m to a finitely additive measure on the Ξ΄-ring generated by . Extensions of finitely additive measures have not been considered so far in the literature. In Section 3 we prove such extension theorems (Theorems 3.6 and 3.7). In Theorem 3.8 and Corollary 3.9 we give conditions under which the extended measure is Οƒ-additive. A particular case of

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