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Journal of the London Mathematical Society 2000 62(1):117-126; doi:10.1112/S0024610700008905
© 2000 by London Mathematical Society
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© The London Mathematical Society

A Constructive Minimal Integral which Includes Lebesgue Integrable Functions and Derivatives

B. Bongiorno, L. Di Piazza and D. Preiss

Dipartimento di Matematica ed Applicazioni Via Archirafi 34, 90123 Palermo, Italy
Department of Mathematics, University College London London, WC1E 6BT

Received 14 December 1998. Revision received 19 April 1999.

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.


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