Let $G$ be a Lie group (maybe LCH and second countable topological group is enough here) and $\mu$ be a Borel probability measure on $G$ (not necessarily Haar). Suppose $U$ is an open subset of $G$ of positive measure. I wonder if the follow assertion is true:
There exists open subsets $V,W$ of positive measure such that $VW\subset U$.
My thoughts: certainly I can find open subsets $V,W$ with $VW\subset U$ by the continuity of the multiplication, but I don't see whether $V,W$ have positive measure.