Completeness of \({\mathcal L}^ p\) spaces and Radon-Nikodym theorems for unbounded finitely additive measures (Q1380311)

The authors characterize completeness of the Lebesgue spaces \({\mathcal L}^p(\Omega,{\mathcal F},\mu)\) for finitely additive measures \(\mu\) on the field \({\mathcal F}\) of subsets of a set \(\Omega\) in the bounded as well as in the unbounded case. Moreover, necessary and sufficient conditions are given by means of Hahn decompositions such that a signed finitely additive measure has a Radon-Nikodým derivative relative to a semidefinite finitely additive measure based on an integral by Maharam.