  In a previous paper of Winter and the author the Law of Large Numbers for the
local mass of certain superdiffusions was proved under a spectral theoretical
assumption, which is equivalent to the ergodicity (positive recurrence) of the
motion component of an $H$-transformed (or weighted) superprocess. In fact the
assumption is also equivalent to the property that the scaling for the
expectation of the local mass is pure exponential.
  In this paper we go beyond ergodicity, that is we consider cases when the
scaling is not purely exponential. Inter alia, we prove the analog of the
Watanabe-Biggins Law of Large Numbers for super-Brownian motion (SBM).
  We will also prove another Law of Large Numbers for a bounded set moving with
subcritical speed, provided the variance term decays sufficiently fast.
