Integral of Map into Abelian $\Omega$-group

Abstract

The common in ring, module and algebra is that they are Abelian group with respect to addition. This property is enough to study integration. I treat integral of measurable map into normed Abelian $\Omega$-group. Theory of integration of maps into $\Omega$-group has a lot of common with theory of integration of functions of real variable. However I had to change some statements, since they implicitly assume either compactness of range or total order in $\Omega$-group.