The Lebesgue Decomposition of any finite, positive, regular Borel measure, $\mu$, on the unit circle in the complex plane with respect to Lebesgue measure can be constructed by considering the intersection of the reproducing kernel Hilbert space of its Cauchy Transforms with the Hardy Space of analytic functions in the complex unit disk. The MacLaurin series coefficients of the analytic $\mu-$Cauchy Transform of the constant function coincide with the moments of the measure $\mu$.\\

Replacing Taylor series indexed by the non-negative integers (the universal singly-generated monoid) by power series indexed by the free (universal) monoid on $d$ generators, we show that the Cauchy Transform approach to Lebesgue Decomposition (and related results including Fatou's Theorem) extends naturally to `non-commutative measures' and Non-commutative reproducing kernel Hilbert spaces of non-commutative functions.

Joint work with Michael T. Jury, University of Florida.