Admissible Translates for Probability Distributions

Abstract

A real number $t$ is an admissible translate of a probability $\varphi$ if $\varphi (A) = 0$ implies that $\varphi_t(A) \equiv \varphi (A - t) = 0$. Conditions are given on its set of admissible translates which ensure that $\varphi$ has a density. The theorems also describe the set where the density is positive and contain as a corollary the result that if $\varphi$ is not absolutely continuous, then the set of admissible translates has an empty interior.