# ANNALES DE L'INSTITUT FOURIER

Closed convex hull of set of measurable functions, Riemann-measurable functions and measurability of translations
Annales de l'Institut Fourier, Tome 32 (1982) no. 1, pp. 39-69.

Soit $G$ un groupe localement compact. Soit ${L}_{t}$ la translation à gauche dans ${L}^{\infty }\left(G\right)$ donnée par ${L}_{t}f\left(x\right)=f\left(tx\right)$. On caractérise (sous des axiomes peu restrictifs de théorie des ensembles) les $f\in {L}^{\infty }\left(G\right)$ telles que l’application $t\to {L}_{t}f$ de $G$ dans ${L}^{\infty }\left(G\right)$ soit scalairement mesurable (c’est-à-dire que $t\to \varphi \left({L}_{t}f\right)$ est mesurable pour $\varphi \in {L}^{\infty }\left(G{\right)}^{*}$). On montre que c’est le cas dès que pour tout caractère $\theta$ de $L\left(G\right)$, $t\to \theta \left({L}_{f}t\right)$ est mesurable, et dans le cas compact, cela caractérise les fonctions Riemann-mesurables. On montre que l’image réciproque de tout borélien de ${L}^{\infty }\left(G\right)$ par l’application $t\to {L}_{t}f$ est mesurable si et seulement si $f$ est uniformément continue.

Les outils de théorie de la mesure utilisés ont un intérêt en soi. Par exemple un ensemble de fonctions mesurables sur $\left[0,1\right]$ est séparable et relativement compact pour la topologie de la convergence ponctuelle, il en est de même de son enveloppe convexe.

Let $G$ be a locally compact group. Let ${L}_{t}$ be the left translation in ${L}^{\infty }\left(G\right)$, given by ${L}_{t}f\left(x\right)=f\left(tx\right)$. We characterize (undre a mild set-theoretical hypothesis) the functions $f\in {L}^{\infty }\left(G\right)$ such that the map $t\to {L}_{t}f$ from $G$ into ${L}^{\infty }\left(G\right)$ is scalarly measurable (i.e. for $\varphi \in {L}^{\infty }\left(G{\right)}^{*}$, $t\to \varphi \left({L}_{t}f\right)$ is measurable). We show that it is the case when $t\to \theta \left({L}_{f}t\right)$ is measurable for each character $\theta$, and if $G$ is compact, if and only if $f$ is Riemann-measurable. We show that $t\to {L}_{t}f$ is Borel measurable if and only if $f$ is left uniformly continuous.

Some of the measure-theoretic tools used there have independent interest. For example, if a set of measurable functions on $\left[0,1\right]$ is separable and point-wise relatively compact, the same is true of its convex hull.

@article{AIF_1982__32_1_39_0,
author = {Talagrand, Michel},
title = {Closed convex hull of set of measurable functions, Riemann-measurable functions and measurability of translations},
journal = {Annales de l'Institut Fourier},
pages = {39--69},
publisher = {Imprimerie Louis-Jean},
volume = {32},
number = {1},
year = {1982},
doi = {10.5802/aif.859},
zbl = {0452.28004},
mrnumber = {83g:28007},
language = {en},
url = {aif.centre-mersenne.org/item/AIF_1982__32_1_39_0/}
}
Talagrand, Michel. Closed convex hull of set of measurable functions, Riemann-measurable functions and measurability of translations. Annales de l'Institut Fourier, Tome 32 (1982) no. 1, pp. 39-69. doi : 10.5802/aif.859. https://aif.centre-mersenne.org/item/AIF_1982__32_1_39_0/

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