  The thermal equilibrium distribution over quantum-mechanical wave functions
is a so-called Gaussian adjusted projected (GAP) measure, $GAP(\rho_\beta)$,
for a thermal density operator $\rho_\beta$ at inverse temperature $\beta$.
More generally, $GAP(\rho)$ is a probability measure on the unit sphere in
Hilbert space for any density operator $\rho$ (i.e., a positive operator with
trace 1). In this note, we collect the mathematical details concerning the
rigorous definition of $GAP(\rho)$ in infinite-dimensional separable Hilbert
spaces. Its existence and uniqueness follows from Prohorov's theorem on the
existence and uniqueness of Gaussian measures in Hilbert spaces with given mean
and covariance. We also give an alternative existence proof. Finally, we give a
proof that $GAP(\rho)$ depends continuously on $\rho$ in the sense that
convergence of $\rho$ in the trace norm implies weak convergence of
$GAP(\rho)$.
