×

Boundedness and convergence theorems in effect algebras. (English) Zbl 1164.28002

Generalizations and analogues of important classical measure-theoretic results such as the Nikodym boundedness and Vitali-Hahn-Saks theorems have recently been obtained for measures \(\mu \: E\to G\) defined on possibly non-Boolean structures \(E\) and taking values in a topological group \(G\). In this paper, the authors study such measures when \(E\) is a so-called effect algebra and \(G\) is an \(\ell \)-group rather than a topological group. (The definition should have stated that an \(\ell \)-group is lattice ordered.)
By employing double sequences called regulators, {A. Boccuto} and {D. Candeloro} [J. Math. Anal. Appl. 265, 170–194 (2002; Zbl 1006.28012)] defined D-convergence and RD-convergence in the \(\ell \)-group \(G\), argued that these notions were more appropriate than (O)-convergence for measure-theoretic purposes, and used them to obtain Nikodym and Vitali-Hahn-Saks type theorems when \(E\) is a Boolean algebra.
Employing regulators and RD-convergence, the authors of the paper are able to prove a Nikodym boundedness type theorem for the more general case, in which \(E\) is a lattice ordered effect algebra (i.e., a D-lattice) and \(\mu \) is a modular measure (i.e., \(\mu (a\vee b)+\mu (a\wedge b)=\mu (a)+\mu (b)\) for all \(a,b\in E\)).
Other classical results involving uniform exhaustivity and absolute continuity are extended to the case in which \(E\) is an effect algebra.

MSC:

28A12 Contents, measures, outer measures, capacities
06C15 Complemented lattices, orthocomplemented lattices and posets

Citations:

Zbl 1006.28012
PDFBibTeX XMLCite