Integration of mappings of half-ordered rings

Abstract

Lebesgue type integrals of mappings f: E → X, where X is a semiordered ring and E ⊂R, over a measure with values lying in X are studied. Usual properties of the integral are proved, including theorems on limits under the integral as well as those on absolute continuity and complete additivity of the (u, v)-integral. A theorem on the existence of a solution of a differential equation with (u, v)-derivatives is proved.