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A counterexample to ``An extension of the Vitali-Hahn-Saks theorem'' and a compactness result


Author: Guy Degla
Journal: Proc. Amer. Math. Soc. 128 (2000), 2553-2559
MSC (2000): Primary 28A33; Secondary 28C15
DOI: https://doi.org/10.1090/S0002-9939-00-05411-3
Published electronically: February 29, 2000
MathSciNet review: 1676360
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Abstract:

We give a counterexample to ``An extension of the Vitali-Hahn-Saks theorem'' and from that highlight the sharp frame within which any attempt to change the version of such an extension should be possible. Lastly a sequential compactness criterion for Radon measures absolutely continuous with respect to a prescribed Radon measure defined on a locally compact separable metric space (taking into account the ideas of Hernandez-Lerma and Lasserre) is proved. The results deal with Radon measures but yield obvious corollaries on real (or vector-valued) Radon measures and so on functions with bounded variation on open subsets of $\mathbf{R}^n$.


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Additional Information

Guy Degla
Affiliation: Internatinal School for Advanced Studies (SISSA-ISAS), Via Beirut 2-4, 34014 Trieste, Italy
Email: degla@sissa.it

DOI: https://doi.org/10.1090/S0002-9939-00-05411-3
Keywords: Radon, Borel, measures, absolute continuity, convergence, lower semi-continuity, differentiation, Lebesgue point, reflexivity
Received by editor(s): October 1, 1998
Published electronically: February 29, 2000
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 2000 American Mathematical Society

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