## Real Analysis Exchange

- Real Anal. Exchange
- Volume 42, Number 2 (2017), 283-310.

### 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.

#### Article information

**Source**

Real Anal. Exchange, Volume 42, Number 2 (2017), 283-310.

**Dates**

First available in Project Euclid: 10 May 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.rae/1525917689

**Digital Object Identifier**

doi:10.14321/realanalexch.42.2.0283

**Mathematical Reviews number (MathSciNet)**

MR3721803

**Zentralblatt MATH identifier**

06870331

**Subjects**

Primary: 26A03: Foundations: limits and generalizations, elementary topology of the line 26A04

Secondary: 26A05

**Keywords**

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

#### Citation

Sayyad, Redouane. The McShane Integral in the Limit. Real Anal. Exchange 42 (2017), no. 2, 283--310. doi:10.14321/realanalexch.42.2.0283. https://projecteuclid.org/euclid.rae/1525917689