Abstract
As a basis for the whole paper we establish an isomorphism between the lattice Ίfls(R) of all abounded monotone ring topologies on a Boolean ring R and a suitable uniform completion of R; it follows that Wls(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. 0. Introduction. The first part of this paper (Chapter 2) deals with an examination of the lattice Ψts(R) of all ^-bounded monotone ring topologies (= FN-topologies) on a Boolean ring R. In (2.2) an isomorphism is established between Wls(R) and the completion of R for the finest ^-bounded monotone ring topology Us on R. From this result we get some consequences (criteria for completeness and metrizability, decomposition theorems for monotone ring topologies) which are also interesting for measure theory and which — as far as they are known in special cases — were before in each case proved with quite different methods. In the second part of this paper (Chapter 3) a decomposition μ — Σa(ΞA μa into an infinite sum is given for an ^-bounded content μ: R -> G defined on a Boolean ring with values in e.g. a complete Hausdorff topological group (content = finitely additive set function); this decomposition includes the usual decomposition theorems as special cases. For an illustration of the method Chapter 3 first deals with the case \A\— 2, i.e. with the Lebesgue decomposition; this again includes as special cases decompositions of a Hewitt-Yosida type, decompositions into an atomless and an atomic content, into a regular and an antiregular content and others. We explain the arising problems with the Lebesgue decomposition. The classic Lebesgue decomposition {μ — λ + v, λ J_w, v < u) of a nonnegative (σ-additive) measure on a σ-algebra rests on a decomposition of the basic set into two disjoint sets of the σ-algebra. The same is still true
Highlights
The first part of this paper (Chapter 2) deals with an examination of the lattice Ψts(R) of all ^-bounded monotone ring topologies ( = FN-topologies) on a Boolean ring R
As a basis for the whole paper we establish an isomorphism between the lattice Ίfls(R) of all abounded monotone ring topologies on a Boolean ring R and a suitable uniform completion of R; it follows that Wls(R) itself is a complete Boolean algebra
From this result we get some consequences which are interesting for measure theory and which — as far as they are known in special cases — were before in each case proved with quite different methods
Summary
As a basis for the whole paper we establish an isomorphism between the lattice Ίfls(R) of all abounded monotone ring topologies on a Boolean ring R and a suitable uniform completion of R; it follows that Wls(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.
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