A Characterization of Singular Measures

Abstract

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