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Uniformities of Fréchet-Nikodým type on Vitali spaces. (English) Zbl 0966.28008
A Vitali space is a common abstraction of Boolean rings and lattice-ordered groups which has many properties in common with both of these structures; in particular, every Vitali space is a lattice with a commutative partial addition which distributes with the lattice operations (disjoint union in the case of a Boolean ring, complete addition in the case of a lattice-ordered group). Vital spaces were introduced by the reviewer [Compos. Math. 54, 51-62 (1985; Zbl 0561.06010)] (see also [Lect. Notes Math. 1320, 300-341 (1988; Zbl 0664.06010)] and [“Jordan decompositions of generalized vector measures” (1989; Zbl 0692.28004)]). They were also studied by C. Constantinescu [Atti Sem. Mat. Fis. Univ. Modena, Suppl. 35, 1-286 (1989; Zbl 0696.46027)] who introduced their name. Part of the interest in this algebraic structure results from the fact that additive functions on a Vitali space generalize vector measures on a Boolean ring and linear operators on a Riesz space.
In the spirit of developing a common theory for measures on Boolean rings and linear operators on Riesz spaces, the present paper introduces and studies Fréchet-Nikodým uniformities on Vitali spaces which make the lattice operations as well as partial addition and partial subtraction uniformly continuous and which are uniquely determined by the system of \(0\)-neighbourhoods. Such uniform Vitali spaces unify topological Boolean rings and locally solid Riesz spaces. Among many other results, the author shows that the class of uniform Vitali spaces is closed under quotients and uniform completion. A second group of results establishes general decomposition theorems of the Lebesgue and Yosida-Hewitt type for additive functions from a Vitali space into a commutative topological group.

MSC:
28B05 Vector-valued set functions, measures and integrals
28B10 Group- or semigroup-valued set functions, measures and integrals
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
06B30 Topological lattices
22A26 Topological semilattices, lattices and applications
46G10 Vector-valued measures and integration
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