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A CONSTRUCTIVE MINIMAL INTEGRAL WHICH INCLUDES LEBESGUE INTEGRABLE FUNCTIONS AND DERIVATIVES

Published online by Cambridge University Press:  30 October 2000

B. BONGIORNO ,
L. DI PIAZZA  and
D. PREISS
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
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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 F – H is absolutely continuous. They strengthen this definition by proving that F – H can have arbitrary small variation. Nevertheless, their definition still needs a choice of a differentiable function.

Type
Research Article
Information
Journal of the London Mathematical Society , Volume 62 , Issue 1 , August 2000 , pp. 117 - 126
Copyright
The London Mathematical Society 2000

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