On concentrated probabilities

Let G be a locally compact Polish group with an invariant metric. We provide sufficient and necessary conditions for the existence of a compact set A ⊆ G and a sequence ${\displaystyle g_n\in G}$ such that ${\displaystyle {\mu }^{\ast n}\left(g_{n}A\right)\equiv 1}$ for all n. It is noticed that such measures μ form a meager subset of all probabilities on G in the weak measure topology. If for some k the convolution power ${\displaystyle {\mu }^{\ast k}}$ has nontrivial absolutely continuous component then a similar characterization is obtained for any locally compact, σ-compact, unimodular, Hausdorff topological group G.