Definition. A subset $K$ of a topological group $X$ is called measure-continuous if there exists a $\sigma$-additive Borel probability measure $\mu$ on $X$ such that for every compact subset $C\subset X$ the map $f:K\to [0,1]$, $f:x\mapsto \mu(C+x)$ is continuous.
Remark 1. Each measure-continuous set $K$ in a Polish group $X$ is contained in a $\sigma$-compact subgroup; so $K$ is small in a sense. What about the converse?
Problem. Is each compact subset of a Banach space $X$ measure-continuous? What is the answer for classical Banach spaces $c_0$ or $\ell_p$?
Remark 2. It can be shown that each compact subset in the countable product of locally compact topological groups is measure-continuous.