## Real Analysis Exchange

### An Integral on a Complete Metric Measure Space

#### Abstract

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

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

Dates
First available in Project Euclid: 1 July 2015

Permanent link to this document
https://projecteuclid.org/euclid.rae/1435759201

Mathematical Reviews number (MathSciNet)
MR3365396

Zentralblatt MATH identifier
06848829

#### Citation

Bongiorno, Donatella; Corrao, Giuseppa. An Integral on a Complete Metric Measure Space. Real Anal. Exchange 40 (2015), no. 1, 157--178. https://projecteuclid.org/euclid.rae/1435759201