I am not very conversant about this topic and was given this question as an assignment. This is my first post so I do apologize if the question is very silly.
Let $\mu$ be any probability measure on a locally compact topological group $G$. Let $\sigma$ be any probability measure on $G$. Then if the sequence of measures $$\frac{1}{n}(\sigma+\mu*\sigma+\cdots+\mu^{n-1}*\sigma)$$ converges to $\nu$ weakly, show that $\nu$ is $\mu$-stationary, i.e. $\mu*\nu=\nu$.