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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

A nonuniform version of the theorem of Radon-Nikodým in the finitely additive case with applications to extensions of finitely additive set functions


Author: D. Plachky
Journal: Proc. Amer. Math. Soc. 113 (1991), 651-654
MSC: Primary 28A20
MathSciNet review: 1077788
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Abstract: For $ \mu ,\nu \in b{a_ + }(\Omega ,\mathfrak{A})$ it is shown that the existence of a net of nonnegative functions $ {f_{\alpha '}}$ that are primitive relative to $ \mathfrak{A}$ and satisfy $ {\lim _\alpha }{\smallint _A}{f_\alpha }d\mu = \nu (A),A \in \mathfrak{A}$, is equivalent to the condition $ \nu \lesssim \mu $, i.e. $ \mu (A) = 0$ for some $ A \in \mathfrak{A}$ implies $ \nu (A) = 0$. Furthermore, as an application it is proved that for $ \mu ,\nu \in b{a_ + }(\Omega ,\mathfrak{A})$ satisfying $ \nu \lesssim \mu $ and any extension $ \mu ' \in b{a_ + }(\Omega ,\mathfrak{A}')$ of $ \mu $, where $ \mathfrak{A}'$ denotes some algebra of subsets of $ \Omega $ containing $ \mathfrak{A}$, there exists some extension $ \nu ' \in b{a_ + }(\Omega ,\mathfrak{A}')$ of $ \nu $ such that $ \nu ' \lesssim \mu '$ is valid.


References [Enhancements On Off] (What's this?)

  • [1] K. P. S. Bhaskara Rao and M. Bhaskara Rao, Theory of charges, Pure and Applied Mathematics, vol. 109, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1983. A study of finitely additive measures; With a foreword by D. M. Stone. MR 751777 (86f:28006)
  • [2] N. Dunford and J. Schwartz, Linear operators, Part I, Interscience, New York, 1964.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1991-1077788-7
PII: S 0002-9939(1991)1077788-7
Keywords: Nonuniform version of the Radon-Nikodym theorem, finitely-additive set functions
Article copyright: © Copyright 1991 American Mathematical Society