Un exemple de fonction int\'egrable sur [-1, 1] mais pas sur [0, 1]

TL;DR

This paper presents an example of a function that is integrable over [-1, 1] but not over [0, 1], illustrating the importance of completeness in the space of values for the Chasles property to hold.

Contribution

It provides a concrete example of a function in an infinite dimensional space that violates the Chasles property and clarifies the role of completeness in this context.

Findings

An explicit example of a function integrable on [-1, 1] but not on [0, 1]

Graphical representation of functions with infinite dimensional values

Chasles' property holds iff the target space is complete

Abstract

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.