Über einen abstrakten Jordaninhalt für Zerlegungsstrukturen. (About an abstract Jordan volume for decomposition structures) (Q1121527)

Let (R,G) be a general Klein space, i.e. R is a nonempty set with a group G of transformations acting on R. For any class \({\mathcal E}\) of subsets of R a function \(\mu\) on \({\mathcal E}\) is called an abstract elementary content if \(\mu\) is a G-invariant finitely additive functional which takes its values in a commutative semigroup. The author gives three axioms for \({\mathcal E}\) such that the set \({\mathcal E}/\sim\) of equivalence classes with respect to G-equidecomposability \(\sim\) forms a commutative semigroup and so the canonical homomorphism of \({\mathcal E}\) onto \({\mathcal E}/\sim\) is an abstract content. Such semigroups of equidecomposability can be found, for example, in \textit{S. Wagon}, ``The Banach-Tarski Paradox'', Cambridge Univ. Press (1985; Zbl 0569.43001). The main result of this paper is the construction of an extension of such an abstract content functional to an abstract Jordan measure with methods of the analysis in ordered commutative semigroups.