  A positive non-commutative (NC) measure is a positive linear functional on
the free disk operator system which is generated by a $d$-tuple of
non-commuting isometries. By introducing the hybrid forms, their Cauchy
transforms, and techniques from NC reproducing kernel Hilbert spaces (RKHS), we
construct a natural Lebesgue decomposition for any positive NC measure against
any other such measure. Our work extends the Jury-Martin decomposition, which
originally decomposes positive NC measures against the standard NC Lebesgue
measure. In fact, we give a more generalized definition of absolute continuity
and singularity, which reduces to their definition when the splitting measure
is the standard NC Lebesgue measure. This generalized definition makes it
possible to extend Jury-Martin theory for any splitting NC measure, and it
recovers their decomposition when the splitting NC measure is the Lebesgue one.
Our work implies a Lebesgue decomposition for representations of the
Cuntz-Toeplitz C*-algebra. Furthermore, our RKHS method gives a new proof of
the classical Lebesgue decomposition when applied to the classical one
dimensional setting, i.e., $d=1$.
