Abstract
Let X be an abelian Polish group. For every analytic Haar-null set A X let T (A) be the set of test measures of A. We show that T (A) is always dense and co- analytic in P (X). We prove that if A is compact then T (A) is G dense, while if A is non-meager then T (A) is meager. We also strengthen a result of Solecki and we show that for every analytic Haar-null set A, there exists a Borel Haar-null set B A such that T (A)n T (B) is meager. Finally, under Martin's Axiom and the negation of Continuum Hypothesis, some results concerning co-analytic sets are derived.
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