The McShane integral in the limit (Q1750322)

The paper introduces McShane integrability in the limit, which is a concept intermediate between the McShane integral and the weak McShane integral. It is defined for functions defined on a \(\sigma\)-finite outer regular Radon measure space into a Banach space. The author shows conditions, under which McShane integrability follows from McSane integrability in the limit. The properties of the newly defined integrability and correspondence with the Pettis integral are shown. Beppo Levi's version convergence theorem for the McShane integral in the limit is also shown. It is shown that the space of McShane integrable in the limit functions equipped with the Pettis norm is not complete.