Random variables with an invariant random shift in compact metrisable abelian groups

The main result of this paper states that for independent random variables $X, Y$ taking values in a compact metrisable abelian group, $X + Y$ has the same distribution as $X$, if and only if there exists a compact subgroup $A$ such that $P(Y\in A)=1$ and $X + a$ has the same distribution as $X$ for all $a\in A$. As a conclusion from the above it is shown that for independent random variables $X, Y$ such that $X+Y$ has the same distribution as $X$, $X+Y$ and $Y$ are also independent. It becomes also apparent that the distribution of $X$ is the Haar measure (uniform distribution) if for each open set $U$, $P(Y\in U) > 0$.