As a basis for the whole paper
we establish an isomorphism between the lattice Ms(R) of all s-bounded monotone
ring topologies on a Boolean ring R and a suitable uniform completion of R; it
follows that Ms(R) itself is a complete Boolean algebra. Using these facts we study
s-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.
