Comparison Between Several Ways of Defining Lebesgue’s Integral After Lebesgue
Keywords:Lebesgue’s integral, Integration without recourse to Lebesgue measure theory, Integral as limit of integrals of elementary functions, Lebesgue’ integral as functional prolongation of Cauchy integral, Abstract integration spaces
After the publication in 1904 of Lebesgue’s Leçons, many mathematicians had the idea of obtaining a simpler and more natural theory of integration, sometimes for a technical or philosophical purpose, sometime for a didactical purpose. Among them Borel, Riesz, Young, Hahn, Pierpoint, Radon, Fréchet, Daniell and in Italy Tonelli, Vitali, Caccioppoli dealt with this fundamental tool. In this paper, their theories are exposed and compared.
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