Quelles fonctions changent toute loi uniforme en une loi uniforme? (Q797894)

Consider a measurable function f:\({\mathbb{R}}\to {\mathbb{R}}\) such that for any interval [a,b], if the random variable X is uniformly distributed on [a,b], then f(X) is uniformly distributed on some interval \([a_ 1,b_ 1]\) (depending on [a,b]). This paper proves that there exist 2 real numbers \(\alpha\) and \(\beta\) such that \(f(x)=\alpha x+\beta\) for almost all x. This result has been stated without proof by \textit{J. Pradines} and the reviewer in their paper in C. R. Acad. Sci., Paris, Ser. A 286, 399- 402 (1978; Zbl 0372.60012). The result is more delicate than it seems and the proof presented here is short and elegant.