Open Access
2017 The McShane Integral in the Limit
Redouane Sayyad
Real Anal. Exchange 42(2): 283-310 (2017). DOI: 10.14321/realanalexch.42.2.0283


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.


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Redouane Sayyad. "The McShane Integral in the Limit." Real Anal. Exchange 42 (2) 283 - 310, 2017.


Published: 2017
First available in Project Euclid: 10 May 2018

zbMATH: 06870331
MathSciNet: MR3721803
Digital Object Identifier: 10.14321/realanalexch.42.2.0283

Primary: 26A03 , 26A04
Secondary: 26A05

Keywords: Generalized McShane partition , Locally \(\tau\)\textrm-upper McShane boundedness , McShane integral , McShane integral in the limit , Pettis integral , Weak McShane integral

Rights: Copyright © 2017 Michigan State University Press

Vol.42 • No. 2 • 2017
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