Is the Haar measure of a product of finite measure and compact, finite? Let $G$ be a locally compact group with Haar measure $ \mu $, $K \subset G$ a compact subset and $ F \subset G $ any subset of finite Haar measure $\mu (F) < \infty $. Is the Haar measure of the product $ \mu(KF) $ finite as well? I know that the compactness of $K$ implies that $\mu(K)<\infty$, and that the above would be true if F was compact (since then $ KF$ would be compact as well).

Take $G:=\Bbb R$ (additive group) with Lebesgue measure, $K:=[0,1]$ and $F:=\Bbb Z$. Then $$KF=\\{x+y,x\in [0,1],y\in\Bbb Z\\}=\Bbb R$$ which has infinite measure.