Journal of Statistical Theory and Practice

, Volume 5, Issue 3, pp 453–473 | Cite as

Radon-Nikodým Derivatives of Hilbert Space Valued Measures

  • Y. KakiharaEmail author


The weak Radon-Nikodým derivative of a Hilbert space valued measure is introduced. We obtain a necessary and sufficient condition for the existence of such a measure and study some of its properties. Integration of a scalar valued function with respect to a Hilbert space valued measure having a weak Radon-Nikodým derivative is seen to be related to the usual Dunford-Schwartz integral. Finally, we examine the case where the measure has values in the class of Hilbert-Schmidt operators.

AMS Subject Classification

28B05 60G12 


Hilbert space valued measures Weak Radon-Nikodým derivatives v-boundedness 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Abreu, J.L., Salehi, H., 1984. Schauder basic measures in Banach and Hilbert spaces. Boletin de la Sociedad Matemática Mexicana, 29, 71–84.MathSciNetzbMATHGoogle Scholar
  2. Diestel, J., Uhl, Jr., J.J., 1977. Vector Measures. American Mathematical Society, Providence, RI.CrossRefGoogle Scholar
  3. Dunford, N., Schwartz, J.T., 1958. Linear Operators, Part I. Interscience, New York.zbMATHGoogle Scholar
  4. Kakihara, Y., 1997. Multidimensional Second Order Stochastic Processes. World Scientific, Singapore.CrossRefGoogle Scholar
  5. Kakihara, Y., 2007. Integration with respect to Hilbert-Schmidt class operator valued measures. Proceedings of 2005 Symposium on Applied Functional Analysis - Information Science and Related Topics, Murofushi, T., Takahashi W. and Tsukada, M. (Editors), Yokohama Publishers, Yokohama, pp. 263–278.Google Scholar
  6. Kakihara, Y., 2010. Dunford-Schwartz type integral and application to Cramér and Karhunen processes. Submitted.Google Scholar
  7. Rao, M.M., 1973. Remarks on a Radon-Nikodým theorem for vector measures. Vector and Operator Valued Measures and Applications, Tucker, D.H. and Maynard, H.B. (Editors), Academic Press, New York, pp.303–317.CrossRefGoogle Scholar
  8. Whitney, H., 1956. Geometric Integration Theory. Princeton University Press, Princeton.zbMATHGoogle Scholar

Copyright information

© Grace Scientific Publishing 2011

Authors and Affiliations

  1. 1.Department of MathematicsCalifornia State UniversitySan BernardinoUSA

Personalised recommendations