  This note explains how the two measures used to define the $\mu$-deformed
Segal-Bargmann space are natural and essentially unique structures. As is well
known, the density with respect to Lebesgue measure of each of these measures
involves a Macdonald function. Our primary result is that these densities are
the solution of a system of ordinary differential equations which is naturally
associated with this theory. We then solve this system and find the known
densities as well as a "spurious" solution which only leads to a trivial
holomorphic Hilbert space. This explains how the Macdonald functions arise in
this theory. Also we comment on why it is plausible that only one measure will
not work. We follow Bargmann's approach by imposing a condition sufficient for
the $\mu$-deformed creation and annihilation operators to be adjoints of each
other. While this note uses elementary techniques, it reveals in a new way
basic aspects of the structure of the $\mu$-deformed Segal-Bargmann space.
