Alexander Schied
\def\cadlag{{c\`{a}dl\`{a}g}}
We extend the class of $(\xi,\psi,K)$-superprocesses known so far by applying a simple transformation induced by a \lq\lq weight function\rq\rq\ for the one-particle motion. These transformed superprocesses may exist under weak conditions on the branching parameters, and their state space automatically extends to a certain space of possibly infinite Radon measures. It turns out that a number of superprocesses which were so far not included in the general theory fall into this class. For instance, we are able to extend the hyperbolic branching catalyst of Fleischmann and Mueller to the case of $\beta$-branching. In the second part of this paper, we discuss regularity properties of our processes. Under the assumption that the one-particle motion is a Hunt process, we show that our superprocesses possess right versions having \cadlag\ paths with respect to a natural topology on the state space. The proof uses an approximation with branching particle systems on Skorohod space.