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A Constructive Minimal Integral which Includes Lebesgue Integrable Functions and Derivatives
Author(s) -
Bongiorno B.,
Di Piazza L.,
Preiss D.
Publication year - 2000
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610700008905
Subject(s) - lebesgue integration , mathematics , riemann integral , differentiable function , pure mathematics , constructive , function (biology) , integrable system , daniell integral , generality , algebra over a field , process (computing) , computer science , fourier integral operator , psychology , evolutionary biology , psychotherapist , biology , operator theory , operating system
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.