Let $G$ be a locally compact abelian group (I am mostly interested in $\mathbb{R}^d$) with Haar measure $\mu$ be the Haar measure. Consider a fixed compact set $K$ containing the identity. Is it true that $$ f(E) = \mu(E + K) / \mu(E) $$ is a bounded function for $E$ ranging in the compact sets? it is possible to do this for $E$ ranging in the open sets? I think this has to do with $G$ being amenable.
Edit: The case I am particularly interested is for sequences $f(E)$ when $E$ is growing to $G$, i.e. $\mu(U) \to \mu(G)$ since I am particularly interested in computing the supremum of such $f$ when $E$ ranges in an increasing Folner sequence.