The McShane Integral in the Limit

Abstract

We introduce the notion of the McShane integral in the limit for functions defined on a \(\sigma\)-finite outer regular quasi Radon measure space \((S,\Sigma,\mathcal{T},\mu)\) into a Banach space \(X\) and we study its relation with the generalized McShane integral introduced by D. H. Fremlin. It is shown that if a function from \(S\) into \(X\) is McShane integrable in the limit on \(S\) and scalarly locally \(\tau\)-upper McShane bounded for some \(\tau >0\), then it is McShane integrable on \(S\). On the other hand, we prove that if an \(X\)-valued function is McShane integrable in the limit on \(S\), then it is McShane integrable on each member of an increasing sequence \((S_\ell)_{\ell\geq 1}\) of measurable sets of finite measure with union \(S\). We also prove a version of Beppo Levi’s theorem for this new integral.