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Précis de mathématiques et de statistiques

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Analyse réelle et complexe, traitement du signal, probabilités et statistiques.
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Summary. It is well known that Riemann sums converge toward the integral of any Riemannintegrable function f on the segment [a, b]. In some cases this property is still valid for generalized integrals and can be useful to solve problems involving series and sums. In this note we prove (Theorem 3) that the result still holds for a generalized integral under the assumption that the function is decreasing on the interval.We give two counterexamples: the first one with a linear piecewise function, and a second one based on an integral suggested by Hardy. The last section shows recent applications of the Riemann sums in computer science and applied mathematics.
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