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2003-2004 A characterization of singular measures.
Vilmos Prokaj
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Real Anal. Exchange 29(2): 805-812 (2003-2004).


Denote by $\mu$ a probability Borel measure on the real line and by $\tau_c$ the translation by $c.$ We show that $\mu$ is singular with respect to Lebesgue measure if and only if the set of those $c$ for which $\mu$ and $\tau_c\mu$ are mutually singular is dense (Theorem 1). Another characterization of singularity (Theorem 10) is the existence of a set of full $\mu$ measure that has continuum many disjoint translates. This result is also linked to some known results about $\sigma$-porous sets


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Vilmos Prokaj. "A characterization of singular measures.." Real Anal. Exchange 29 (2) 805 - 812, 2003-2004.


Published: 2003-2004
First available in Project Euclid: 7 June 2006

zbMATH: 1062.28002
MathSciNet: MR2083815

Primary: 28A05 , 28A12 , 28A35

Keywords: $\sigma$-porous set , singular measure

Rights: Copyright © 2003 Michigan State University Press

Vol.29 • No. 2 • 2003-2004
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