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Proceedings of the American Mathematical Society

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

 

 

Convolutions of continuous measures and sums of an independent set


Author: James Michael Rago
Journal: Proc. Amer. Math. Soc. 44 (1974), 123-128
MSC: Primary 43A05
DOI: https://doi.org/10.1090/S0002-9939-1974-0330921-4
MathSciNet review: 0330921
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Abstract: Let $ E$ be a compact independent subset of an l.c.a. group $ G;{\mu _1}, \cdots ,{\mu _{n + 1}}$ continuous regular bounded Borel measures on $ G$; and $ {k_1}, \cdots ,{k_n}$ integers. Let $ {k_i} \times E = \{ {k_i}x\vert x \in E\} $. We prove (1) $ {\mu _1} \ast \cdots \ast {\mu _{n + 1}}({k_1} \times E + \cdots + {k_n} \times E) = 0$ (the proof is a combinatorial argument).

As a corollary of (1) we obtain (2) if $ H$ is any closed nondiscrete subgroup of $ G$, then the intersection of $ H$ with the group generated by $ E$ has zero $ H$-Haar measure.


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DOI: https://doi.org/10.1090/S0002-9939-1974-0330921-4
Keywords: Compact independent set, Haar measure
Article copyright: © Copyright 1974 American Mathematical Society