Strong law of large number for branching Hunt processes

In this paper we prove that, under certain conditions, a strong law of large number holds for a class of branching particle systems $X$ corresponding to the parameters $(Y,\beta,\psi)$, where $Y$ is a Hunt process and $\psi$ is the generating function for the offspring. The main tool of this paper is the spine decomposition and we only need a $L\log L$ condition.