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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Convolutions and absolute continuity


Author: D. A. Lind
Journal: Proc. Amer. Math. Soc. 39 (1973), 347-348
MSC: Primary 28A10; Secondary 43A05
DOI: https://doi.org/10.1090/S0002-9939-1973-0320257-9
MathSciNet review: 0320257
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Abstract: We show that if $ E$ is a subset of the circle with positive Lebesgue measure, and $ g$ is integrable on almost every translate of $ E$, then $ g$ is integrable on the whole circle. A generalization of this fact leads to a characterization of positive measures with nonvanishing absolutely continuous part.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0320257-9
Keywords: Compact group, integrable on translates, positive measure, singular measure, finiteness of convolutions, absolute continuity
Article copyright: © Copyright 1973 American Mathematical Society