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Archiv der Mathematik

, Volume 97, Issue 6, pp 559–567 | Cite as

Fibers of the L algebra and disintegration of measures

  • Marek Kosiek
  • Krzysztof Rudol
Open Access
Article

Abstract

It is shown that Gelfand transforms of elements \({f\in L^{\infty} (\mu)}\) are almost constant at almost every fiber \({\Pi^{-1}(\{x\})}\) of the spectrum of L (μ) in the following sense: for each \({f\in L^{\infty} (\mu)}\) there is an open dense subset U = U(f) of this spectrum having full measure and such that the Gelfand transform of f is constant on the intersection \({\Pi^{-1}(\{x\})\cap U}\). As an application a new approach to disintegration of measures is presented, allowing one to drop the usually taken separability assumption.

Mathematics Subject Classification (2010)

Primary 46J10 28A50 Secondary 46E30 28A20 

Keywords

Function algebra Measure L algebra Fiber Disintegration 

Notes

Acknowledgments

The authors wish to thank Professors Christian Berg, Jan Stochel and Edward Tutaj for valuable remarks.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

References

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    Bourbaki N.: Élements de mathématique. Livre VI, Intégration. Hermann, Paris (1959)Google Scholar
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    Dixmier J.: Sur certain espaces considérés par M. H. Stone. Summa Brasil. Math. 2, 151–182 (1951)MathSciNetGoogle Scholar
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    Gamelin T.W.: Uniform Algebras. Prentice Hall, Inc.,, Englewood Clifs, N.J (1969)MATHGoogle Scholar

Copyright information

© The Author(s) 2011

Authors and Affiliations

  1. 1.Instytut MatematykiUniwersytet JagiellońskiKrakówPoland
  2. 2.Wydział Matematyki StosowanejAGH University of Science and TechnologyKrakówPoland

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