  In this paper we prove that, under certain conditions, a strong law of large
numbers holds for a class of super-diffusions $X$ corresponding to the
evolution equation $\partial_t u_t=L u_t+\beta u_t-\psi(u_t)$ on a bounded
domain $D$ in $\R^d$, where $L$ is the generator of the underlying diffusion
and the branching mechanism
  $\psi(x,\lambda)=1/2\alpha(x)\lambda^2+\int_0^\infty (e^{-\lambda
r}-1+\lambda r)n(x, {\rm d}r)$ satisfies $\sup_{x\in D}\int_0^\infty (r\wedge
r^2) n(x,{\rm d}r)<\infty$.
