If $\mu$ and $\nu$ are two positive measures on a compact topological group $G$ such that $0\leq \nu\leq \mu$. Assume $h$ is a probability measure such that $\mu*h=h*\mu=\mu(G)h$, then $\nu*h=h*\nu=\nu(G)h$.
Where $*$ is convolution of measures. Such an lemma is used in proof of Haar state on any compact Quantm gorup. I just want to see how we can prove this in case of $C(G)$.