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) 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 $\varphi \in L^{\infty }(G{)}^{*}$). 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 $[0,1]$ 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 $\varphi \in L^{\infty }(G{)}^{*}$, $t\to \varphi \left(L_{t}f\right)$ 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 $[0,1]$ is separable and point-wise relatively compact, the same is true of its convex hull.

[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 | Zbl

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

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

[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 | Zbl

[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

[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 | Zbl

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

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

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