ABSTRACT
This chapter provides a nonstandard treatment to the problem of extending a tight content on the compact sets of a topological space to a Borel measure. The idea of applying the Loeb measure construction to abstract measure spaces has served as motivation to define monads in terms of arbitrary lattices of sets, rather than topologies. Since lattices are closed under finite unions and finite intersections, saturation arguments will apply. Furthermore, by considering the topology generated by any such lattice, the usual results in the topological case also hold. The basic difference between using an arbitrary lattice on which a set function is defined, and a topology, is that in the first case one considers which properties the set function must satisfy in order to obtain representation results, while in the second case one considers topological properties such as metrizability, regularity, and the like.
