Topological Boolean rings. Decomposition of finitely additive set functions

Abstract

As a basis for the whole paper we establish an isomorphism between the lattice fl s (R) of all abounded monotone ring topologies on a Boolean ring R and a suitable uniform completion of R; it follows that Wl s (R) itself is a complete Boolean algebra. Using these facts we study 5-bounded monotone ring topologies and topological Boolean rings (conditions for completeness and metriziability, decompositions). In the second part of this paper we give a simple proof of a Lebesgue-type decomposition for finitely additive (e.g. semigroup-valued) set functions on a ring, which was first proved by Traynor (in the group-valued case) answering a question of Drewnowski. Using the Lebesgue-decomposition various other decompositions are obtained.