We introduce an abstract treatment of the weak convergence for bounded monotone set functions which allows us to obtain some basic results generalizing well known theorems regarding classical weak and vague convergence and weak convergence of masses on normal topological spaces (e.g. Portmanteau type theorem, Direct and Converse Prokhorov type theorem). Moreover, we introduce a suitable topology (called the Lévy-topology) in order to study the properties of this abstract convergence from a topological point of view.
