Dissipation of Convolution Powers in a Metric Group

In contrast to what is known about probability measures on locally compact groups, a metric group G can support a probability measure μ which is not carried on a compact subgroup but for which there exists a compact subset C⊆G such that the sequence μn(C) fails to converge to zero as n tends to ∞. A noncompact metric group can also support a probability measure μ such that supp μ=G and the concentration functions of μ do not converge to zero. We derive a number of conditions which guarantee that the concentration functions in a metric group G converge to zero, and obtain a sufficient and necessary condition in order that a probability measure μ on G satisfy lim n→∞μn(C)=0 for every compact subset C⊆G.