Vol. 110, No. 2, 1984

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Topological Boolean rings. Decomposition of finitely additive set functions

Hans Weber

Vol. 110 (1984), No. 2, 471–495

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.

Mathematical Subject Classification 2000
Primary: 28B20
Secondary: 06E20
Received: 11 June 1982
Published: 1 February 1984
Hans Weber