Un exemple de fonction intégrable sur [-1, 1] mais pas sur [0, 1]

Details Un exemple de fonction intégrable sur [-1, 1] mais pas sur [0, 1] Un exemple de fonction intégrable sur [-1, 1] mais pas sur [0, 1]

2012-03-23

arxiv.org/abs/1203.5263v1

Mathematics

Functional Analysis

Scientific paper

In courses on integration theory, Chasles property is usually considered as elementary and so "natural" that this is sometimes left to the reader. When the functions take their values in finite dimensional spaces, the property is always verified, but it no more true in infinite dimensional spaces. We first give an easy-to-understand example of a function f from [-1, 1] into the space of polynomial functions from [0, 1] to R which is integrable on [-1, 1] but not on [0, 1]. We also provide a way of representing graphically such a function which explains what means the integral of a function with values in an infinite dimensional space. Then we show that Chasles'property is true if and only if the space in which the functions to integrate take their values is a complete space.

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