Classical Limit Theorems in Lattice Groups

Abstract

We consider several versions of limit theorems for lattice group-valued measures, in which both pointwise convergence of the involved measures and the notions of σ-additivity, (s)-boundedness, regularity, are given in the global sense, that is with respect to a common regulator. We present the construction of some kinds of integrals in the vector lattice context and some Vitali and Lebesgue theorems. Successively we prove some other kinds of limit theorems, in which the main properties of the measures are considered in the classical like sense. Finally, we give different types of decomposition theorems for lattice group-valued measures.

Keywords: Axiomatic convergence, Bochner integral, Brooks-Jewett theorem, convergence in L1, convergence in measure, Dieudonné theorem, dominated convergence theorem, Lattice group, Lebesgue decomposition, Nikodým convergence theorem, optimal integral, Rickart integral, Schur theorem, Sobczyk- Hammer decomposition, Stone Isomorphism technique, ultrafilter measure, uniform integrability, Vitali theorem, Vitali-Hahn-Saks theorem, Yosida-Hewitt decomposition.