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)



Limitations on the extendibility of the Radon-Nikodym Theorem

Author: Gerd Zeibig
Journal: Proc. Amer. Math. Soc. 131 (2003), 2491-2500
MSC (2000): Primary 46B22; Secondary 46J10, 46E30
Published electronically: March 11, 2003
MathSciNet review: 1974647
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Given two locally compact spaces $X,Y$ and a continuous map $r: Y \rightarrow X$ the Banach lattice $\text{\normalsize {$\mathcal{C}$ }}_0(Y)$is naturally a $\text{\normalsize {$\mathcal{C}$ }}_0(X)$-module. Following the Bourbaki approach to integration we define generalized measures as $\text{\normalsize {$\mathcal{C}$ }}_0(X)$-linear functionals $\mu : \text{\normalsize {$\mathcal{C}$ }}_0(Y) \rightarrow \text{\normalsize {$\mathcal{C}$ }}_0(X)$. The construction of an $L^1(\mu)$-space and the concepts of absolute continuity and density still make sense. However we exhibit a counter-example to the natural generalization of the Radon-Nikodym Theorem in this context.

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

  • 1. David P. Blecher, Paul S. Muhly, Vern I. Paulsen, Categories of Operator Modules (Morita Equivalence and Projective Modules), Memoirs of the American Mathematical Society 143 (2000). MR 2000j:46132
  • 2. Johann Cigler, Viktor Losert, Peter Michor, Banach Modules and Functors on Categories of Banach Spaces, Lecture Notes in Pure and Applied Mathematics 46 (1979). MR 80j:46112
  • 3. Joseph Diestel, John J. Uhl, Vector Measures, American Mathematical Society, Mathematical Surveys 15 (1977). MR 56:2216
  • 4. Gerald B. Folland, Real Analysis, John Wiley & Sons, Inc. (1999). MR 2000c:00001
  • 5. Paul Malliavin, Integration and Probability, Graduate Texts in Mathematics 157, Springer Verlag (1995). MR 97f:28001a
  • 6. Helmut H. Schaefer, Banach Lattices and Positive Operators, Springer Verlag (1974). MR 54:11023
  • 7. Gerd Zeibig, Generalized $L^p(\mu)$-spaces, to appear.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 46B22, 46J10, 46E30

Retrieve articles in all journals with MSC (2000): 46B22, 46J10, 46E30

Additional Information

Gerd Zeibig
Affiliation: Department of Mathematical Sciences, Kent State University, Kent, Ohio 44240

Keywords: Banach module, Radon-Nikodym Theorem, Riesz Theorem
Received by editor(s): March 20, 2002
Published electronically: March 11, 2003
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society