Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A reformulation of the Radon-Nikodým theorem


Authors: Jonathan Lewin and Mirit Lewin
Journal: Proc. Amer. Math. Soc. 47 (1975), 393-400
MSC: Primary 28A15
DOI: https://doi.org/10.1090/S0002-9939-1975-0376999-4
MathSciNet review: 0376999
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The Radon-Nikodym theorems of Segal and Zaanen are principally concerned with the classification of those measures $ \mu $ for which any $ \lambda \ll \mu $ is given in the form

$\displaystyle ({\text{i}})\quad \lambda (A) = \int_A {gd\mu } $

for all sets $ A$ of finite $ \mu $ measure.

This paper is concerned with the characterization of those pairs $ \lambda ,\mu $ for which the equality (i) holds for every measurable set $ A$, and introduces a notion of compatibility that essentially solves this problem. In addition, some applications are made to Radon-Nikodym theorems for regular Borel measures.


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

  • [1] P. R. Halmos, Measure theory, Van Nostrand, Princeton, N. J., 1950. MR 11, 504. MR 0033869 (11:504d)
  • [2] E. Hewitt and K. A. Ross, Abstract harmonic analysis. Vol. I: Structure of topological groups. Integration theory, group representations, Die Grundlehren der math. Wissenschaften, Band 115, Academic Press, New York, Springer-Verlag, Berlin, 1963. MR 28 #158. MR 551496 (81k:43001)
  • [3] E. Hewitt and K. Stromberg, Real and abstract analysis. A modern treatment of the theory of functions of a real variable, Springer-Verlag, New York, 1965. MR 32 #5826. MR 0188387 (32:5826)
  • [4] H. L. Royden, Real analysis, Macmillan, New York, 1968. MR 0151555 (27:1540)
  • [5] Walter Rudin, Real and complex analysis, McGraw-Hill, New York, 1966. MR 35 #1420. MR 0210528 (35:1420)
  • [6] I. E. Segal, Equivalences of measure spaces, Amer. J. Math. 73 (1951), 275-313. MR 12, 809. MR 0041191 (12:809f)
  • [7] A. C. Zaanen, The Radon-Nikodym theorem. I, II, Nederl. Akad. Wetensch. Proc. Ser. A 64 = Indag. Math. 23 (1961), 157-187. MR 26 #3862. MR 0146340 (26:3862)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 28A15

Retrieve articles in all journals with MSC: 28A15


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1975-0376999-4
Keywords: Radon-Nikodym theorem, measure space, Borel measure, absolute continuity, differentiation of measures
Article copyright: © Copyright 1975 American Mathematical Society

American Mathematical Society