  In 1976, Donald Cartwright and John McMullen characterized axiomatically the
Riemann-Liouvile fractional integral in a paper that was published in 1978. The
motivation for their work was to answer affirmatively to a conjecture stated by
J. S. Lew a few years before, in 1972. Essentially, their ``Cartwright-McMullen
theorem in fractional calculus'' proved that the Riemann-Liouville fractional
integral is the only continuous extension of the usual integral operator to
positive real orders, in such a way that the Index Law holds. In this paper, we
propose an analogous result for the uniqueness of the extension of the
Stieltjes integral operator, in the case of a smooth integrator.
