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.
