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Group- and vector-valued s-bounded contents. (English) Zbl 0552.28011

Measure theory, Proc. Conf., Oberwolfach 1983, Lect. Notes Math. 1089, 181-198 (1984).
[For the entire collection see Zbl 0539.00008.]
This paper is essentially a unification of parts of my papers which appeared in Stud. Math. 74, 57-81 (1982; Zbl 0445.28008), Commentarii Math. Univ. St. Pauli 31, 49-60 (1982; Zbl 0485.28004), Pac. J. Math. 110, 471-495 (1984; Zbl 0489.28008), here a new method using FN- topologies in measure theory is presented to get a series of old and new results in a unique and comparatively simple way.
The main idea is the following: In order to study s-bounded contents \(\mu,\nu\),... on a Boolean ring R, the continuous extensions \({\tilde \mu},{\tilde \nu}\),... on the uniform completion \((\tilde R,\tilde u)\) of \((R,u)\) with respect to a suitable FN-topology u are first considered; results for \({\tilde \mu}\),\({\tilde \nu}\),... then yield results for the restrictions \(\mu ={\tilde \mu}| R\), \(\nu ={\tilde \nu}| R,...\). Here \(\tilde R\) is an (as lattice) complete Boolean algebra and \({\tilde \mu}\) completely additive; so it is understandable that the examination of \({\tilde \mu}\) is easier than that of \(\mu\). As the base of the sketched completion method, isomorphisms are established between the lattice \({\mathfrak M}_ s(R)\) of all s-bounded FN-topologies on R, a certain sublattice of \({\mathfrak M}_ s(\tilde R)\) and \(\tilde R.\) These isomorphisms give also a deeper insight into the structure of \({\mathfrak M}_ s(R)\).

MSC:

28B10 Group- or semigroup-valued set functions, measures and integrals
28A60 Measures on Boolean rings, measure algebras