A non-commutative version of the first alexandroff decomposition theorem in ordered topological groups

In this paper we establish a decomposition theorem for a positive regular measure on an orthoalgebra with values in an ordered topological group not necessarily commutative. We deduce from it the A. D. Alexandroff’s classical first decomposition theorem and we discuss its uniqueness in the setting of metric spaces. This research is partially supported by the project Analisi Reale of Ministero dell’Università e della Ricerca Scientifica e Tecnologica (Italy) and by Gruppo Nazionale per l’Analisi Funzionale e Applicazioni (Italy).