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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
DOI: https://doi.org/10.1090/S0002-9939-03-07046-1
Published electronically: March 11, 2003
MathSciNet review: 1974647
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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.


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Additional Information

Gerd Zeibig
Affiliation: Department of Mathematical Sciences, Kent State University, Kent, Ohio 44240
Email: gzeibig@math.kent.edu

DOI: https://doi.org/10.1090/S0002-9939-03-07046-1
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

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