  This paper features a comparison inequality for the densities of the moment
measures of super-Brownian motion. These densities are defined recursively for
each $n \ge 1$ in terms of the Poisson and Green's kernels, hence can be
analyzed using the techniques of classical potential theory. When $n = 1$, the
moment density is equal to the Poisson kernel, and the comparison is simply the
classical inequality of Harnack. For $n > 1$ we find that the constant in the
comparison inequality grows at most exponentially with $n$. We apply this to a
class of $X$-harmonic functions $H^\nu$ of super-Brownian motion, introduced by
Dynkin. We show that for a.e. $H^\nu$ in this class, $H^\nu(\mu)<\infty$ for
every $\mu$.
