A Khintchine Decomposition for Free Probability

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

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