Let $\mu$ be a probability measure on the real line. In this paper we prove that there exists a decomposition $\mu = \mu_{ 0} \boxplus \mu_{ 1} \boxplus \... \boxplus \mu_{ n} \boxplus \...$ such that $\mu_{ 0}$ is infinitely divisible and $\mu_{ i}$ is indecomposable for $i \geq 1$. Additionally, we prove that the family of all $\boxplus$-divisors of a measure $\mu$ is compact up to translation. Analogous results are also proven in the case of multiplicative convolution.
Categories: math.OA math.PR