Real Analysis Exchange

An Integral on a Complete Metric Measure Space

Donatella Bongiorno and Giuseppa Corrao

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We study a Henstock-Kurzweil type integral defined on a complete metric measure space \(X\) endowed with a Radon measure \(\mu\) and with a family of “cells” \(\mathcal{F}\) that satisfies the Vitali covering theorem with respect to \(\mu\). This integral encloses, in particular, the classical Henstock-Kurzweil integral on the real line, the dyadic Henstock-Kurzweil integral, the Mawhin’s integral [19], and the \(s\)-HK integral [4]. The main result of this paper is the extension of the usual descriptive characterizations of the Henstock-Kurzweil integral on the real line, in terms of \(ACG^*\) functions (Main Theorem 1) and in terms of variational measures (Main Theorem 2).

Article information

Real Anal. Exchange, Volume 40, Number 1 (2015), 157-178.

First available in Project Euclid: 1 July 2015

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26A39: Denjoy and Perron integrals, other special integrals
Secondary: 28A12: Contents, measures, outer measures, capacities

HK-integral \(ACG^\bigtriangleup\) function critical variation


Bongiorno, Donatella; Corrao, Giuseppa. An Integral on a Complete Metric Measure Space. Real Anal. Exchange 40 (2015), no. 1, 157--178.

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