Disjoint Variation, (s)-Boundedness and Brooks-Jewett Theorems for Lattice Group-Valued k-Triangular Set Functions

摘要

We consider some basic properties of the disjoint variation of lattice group-valued set functions and (s)-boundedness for k-triangular set functions, not necessarily finitely additive or monotone. Using the Maeda-Ogasawara-Vulikh representation theorem of lattice groups as subgroups of continuous functions, we prove a Brooks-Jewett-type theorem for k-triangular lattice group-valued set functions, in which (s)-boundedness is intended in the classical like sense, and not necessarily with respect to a single order sequence. To this aim, we deal with the disjoint variation of a lattice group-valued set function and study the basic properties of the set functions of bounded disjoint variation. Furthermore we show that our setting includes the finitely additive case.