A Note on the Irreducibility of Lebesgue Measure with Applications to Random Walks on the Unit Circle

Let $\mu$ be a probability measure on $R$. We say that a $\sigma$-finite measure $\lambda$ is irreducible with respect to $\mu$ if there does not exist a Borel set $A$ with $\mu (A)$, $\mu ({A^c}) > 0$ such that $\int _A {\mu ({A^c} - x)} \lambda (dx) = 0$. It is well known that the Lebesgue measure $m(dx)$ is irreducible with respect to any discrete measure whose support is $R$. We prove that every absolutely continuous measure is irreducible with respect to any probability measure whose support is $R$ and give an application of this fact to random walks on the unit circle.