An example of a nonzero σ-finite Borel measure μ with everywhere dense linear manifold of admissible (in the sense of invariance) translation vectors is constructed in the Hilbert space ℓ₂ such that μ and any shift of μ by a vector are neither equivalent nor orthogonal. This extends a result established in [7].
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 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)∖ T(B) is meager. Finally, under Martin’s Axiom and the negation of Continuum Hypothesis,...