Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-09T08:08:24.444Z Has data issue: false hasContentIssue false

A CONSTRUCTIVE MINIMAL INTEGRAL WHICH INCLUDES LEBESGUE INTEGRABLE FUNCTIONS AND DERIVATIVES

Published online by Cambridge University Press:  30 October 2000

B. BONGIORNO
Affiliation:
Dipartimento di Matematica ed Applicazioni, Via Archirafi 34, 90123 Palermo, Italy
L. DI PIAZZA
Affiliation:
Dipartimento di Matematica ed Applicazioni, Via Archirafi 34, 90123 Palermo, Italy
D. PREISS
Affiliation:
Department of Mathematics, University College London, London, WC1E 6BT
Get access

Abstract

In this paper we provide a minimal constructive integration process of Riemann type which includes the Lebesgue integral and also integrates the derivatives of differentiable functions. We provide a new solution to the classical problem of recovering a function from its derivative by integration, which, unlike the solution provided by Denjoy, Perron and many others, does not possess the generality which is not needed for this purpose.

The descriptive version of the problem was treated by A. M. Bruckner, R. J. Fleissner and J. Foran in [2]. Their approach was based on the trivial observation that for the required minimal integral, a function F is the indefinite integral of f if and only if F' = f almost everywhere and there exists a differentiable function H such that FH is absolutely continuous. They strengthen this definition by proving that FH can have arbitrary small variation. Nevertheless, their definition still needs a choice of a differentiable function.

Type
Research Article
Copyright
The London Mathematical Society 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)