Avallone, Anna; Rinauro, Silvana; Vitolo, Paolo Boundedness and convergence theorems in effect algebras. (English) Zbl 1164.28002 Tatra Mt. Math. Publ. 35, 159-174 (2007). 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. Reviewer: David J. Foulis (Amherst) Cited in 2 Documents MSC: 28A12 Contents, measures, outer measures, capacities 06C15 Complemented lattices, orthocomplemented lattices and posets Keywords:uniform boundedness theorem; Vitali-Hahn-Saks theorem; absolute continuity; uniform exhaustivity; effect algebra; \(D\)-lattices; measures; \(\ell \)-groups Citations:Zbl 1006.28012 PDFBibTeX XMLCite \textit{A. Avallone} et al., Tatra Mt. Math. Publ. 35, 159--174 (2007; Zbl 1164.28002)