Modules of families of vector measures on a Riemann surface

We consider a Riemann surface in the broad sense of the term in the Hurwitz–Courant terminology and an open set with a compact closure on this surface. In this paper, it is established that a family of vector measures can be associated with a condenser on a given open set, following the Aikawa–Ohtsuka, whose modules are calculated directly with the help of a weighted capacity (with Muckenhoupt weight) of the given condenser.