Closed convex hull of set of measurable functions, Riemann-measurable functions and measurability of translations

Talagrand, Michel

Annales de l'Institut Fourier, Tome 32 (1982), p. 39-69 / Harvested from Numdam

Résumé

Soit $G$ un groupe localement compact. Soit $L_t$ la translation à gauche dans $L^\infty (G)$ donnée par $L_tf(x) = f(tx)$. On caractérise (sous des axiomes peu restrictifs de théorie des ensembles) les $f\in L^\infty (G)$ telles que l’application $t\rightarrow L_tf$ de $G$ dans $L^\infty (G)$ soit scalairement mesurable (c’est-à-dire que $t\rightarrow \varphi (L_tf)$ est mesurable pour $\varphi \in L^\infty (G)^*$). On montre que c’est le cas dès que pour tout caractère $\theta $ de $L(G)$, $t\rightarrow \theta (L_ft)$ est mesurable, et dans le cas compact, cela caractérise les fonctions Riemann-mesurables. On montre que l’image réciproque de tout borélien de $L^\infty (G)$ par l’application $t \rightarrow L_tf$ 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 $[0,1]$ est séparable et relativement compact pour la topologie de la convergence ponctuelle, il en est de même de son enveloppe convexe.

Abstract

Let $G$ be a locally compact group. Let $L_t$ be the left translation in $L^\infty (G)$, given by $L_tf(x) = f(tx)$. We characterize (undre a mild set-theoretical hypothesis) the functions $f\in L^\infty (G)$ such that the map $t\rightarrow L_tf$ from $G$ into $L^\infty (G)$ is scalarly measurable (i.e. for $\varphi \in L^\infty (G)^*$, $t\rightarrow \varphi (L_tf)$ is measurable). We show that it is the case when $t\rightarrow \theta (L_ft)$ is measurable for each character $\theta $, and if $G$ is compact, if and only if $f$ is Riemann-measurable. We show that $t \rightarrow L_tf$ 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 $[0,1]$ is separable and point-wise relatively compact, the same is true of its convex hull.

Détail

Publié le : 1982-01-01

MR 83g:28007 | Zbl 0452.28004

DOI : https://doi.org/10.5802/aif.859

Bibliographie

[1] J. Bourgain, D.H. Fremlin, M. Talagrand, Pointwise compact sets of Baire-measurable functions, American J. of Math., 100, No. 4 (1978), 845-886. | MR 80b:54017 | Zbl 0413.54016

[2] D.H. Fremlin, Pointwise compact sets of measurable functions, Manuscripta Math., 15 (1975), 219-242. | MR 51 #8801 | Zbl 0303.28006

[3] D.H. Fremlin, Measurable functions and almost continuous functions, Manuscripta Math., (to appear). | Zbl 0459.28010

[4] D.H. Fremlin, M. Talagrand, A decomposition theorem for additive set-functions, with applications to Pettis integrals and ergodic means, Math. Z., 168 (1979), 117-142. | MR 80k:28004 | Zbl 0393.28005

[5] A. and C. Ionescu-Tulcea, Topics in the Theory of Liftings, Springer Verlag, 1979.

[6] A. and C. Ionescu-Tulcea, On the existence of a lifting commuting with the left translations of an arbitrary locally compact group, Proceedings Fifth Berkeley Symposium on Math. and Probability, 63-97. | Zbl 0201.49202

[7] J.M. Rosenblatt, Invariant means for the bounded measurable functions on a non-discrete locally compact group, Math. Annalen, 220 (1976), 219-228. | MR 53 #1164 | Zbl 0305.43002

[8] W. Rudin, Homomorphisms and translations in L∞(G), Advances in Math., 16 (1975), 72-90. | MR 51 #3797 | Zbl 0297.22009

[9] J.R. Schoenfield, Martin's axiom, Amer. Math Monthly, 82 (1975), 610-617. | MR 51 #10087 | Zbl 0314.02069

[10] B. Wells, Homomorphisms and translates of bounded functions, Duke Math. J., 41 (1974), 35-39. | MR 49 #1014 | Zbl 0281.28004