On the Additive Property of Finitely Additive Measures

Abstract

By additive property, we refer to a condition under which \(L^p\) spaces over finitely additive measures are complete. In their 2000 paper, Basile and Rao gave a necessary and sufficient condition that a finite sum of finitely additive measures has the additive property. We generalize this result to the case of a countable sum of finitely additive measures. We also apply this result to density measures, the finitely additive probabilities on \(\mathbb {N}\) which extend asymptotic density (also called natural density), and provide the necessary and sufficient condition that a certain type of density measure has the additive property.