### A characterization of algebraic measures

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Let G be a connected locally compact group with a left invariant Haar measure μ. We prove that the function ξ(x) = inf μ̅(AB): μ(A) = x is concave for any fixed bounded set B ⊂ G. This is used to give a new proof of Kemperman’s inequality $\mu \u0332\left(AB\right)\ge min\left(\mu \u0332\right(A)+\mu \u0332(B),\mu (G\left)\right)$ for unimodular G.

We define a class of measures having the following properties: (1) the measures are supported on self-similar fractal subsets of the unit cube ${I}^{M}={[0,1)}^{M}$, with 0 and 1 identified as necessary; (2) the measures are singular with respect to normalized Lebesgue measure m on ${I}^{M}$; (3) the measures have the convolution property that $\mu \ast {L}^{p}\subseteq {L}^{p+\epsilon}$ for some ε = ε(p) > 0 and all p ∈ (1,∞). We will show that if (1/p,1/q) lies in the triangle with vertices (0,0), (1,1) and (1/2,1/3), then $\mu \ast {L}^{p}\subseteq {L}^{q}$ for any measure μ in our class.

We show that in every Polish, abelian, non-locally compact group G there exist non-Haar null sets A and B such that the set {g ∈ G; (g+A) ∩ B is non-Haar null} is empty. This answers a question posed by Christensen.