  The Riesz-Markov theorem identifies any positive, finite, and regular Borel
measure on the complex unit circle with a positive linear functional on the
continuous functions. By the Weierstrass approximation theorem, the continuous
functions are obtained as the norm closure of the Disk Algebra and its
conjugates. Here, the Disk Algebra can be viewed as the unital norm-closed
operator algebra of the shift operator on the Hardy Space, $H^2$ of the disk.
  Replacing square-summable Taylor series indexed by the non-negative integers,
i.e. $H^2$ of the disk, with square-summable power series indexed by the free
(universal) monoid on $d$ generators, we show that the concepts of absolutely
continuity and singularity of measures, Lebesgue Decomposition and related
results have faithful extensions to the setting of `non-commutative measures'
defined as positive linear functionals on a non-commutative multi-variable
`Disk Algebra' and its conjugates.
