  Let $X$ be the countable product of Abelian locally compact Polish groups and
$A,B\subset X$ be two Borel sets, which are not Haar-null in $X$. We prove that
the sum-set $A+B:=\{a+b:a\in A,\;\;b\in B\}$ is Haar-open in the sense that for
any non-empty compact subset $K\subset X$ and point $p\in K$ there exists a
point $x\in X$ such that the set $K\cap(A+B+x)$ is a neighborhood of $p$ in
$K$. This is a generalization of the classical Steinhaus Theorem (1920) to
non-locally compact groups. We do not know if this generalization holds for
Banach spaces.
