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

Let G be a locally compact group. Let L t be the left translation in L (G), given by L t f(x)=f(tx). We characterize (undre a mild set-theoretical hypothesis) the functions fL (G) such that the map tL t f from G into L (G) is scalarly measurable (i.e. for ϕL (G) * , tϕ(L t f) is measurable). We show that it is the case when tθ(L f t) is measurable for each character θ, and if G is compact, if and only if f is Riemann-measurable. We show that tL 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.

Soit G un groupe localement compact. Soit L t la translation à gauche dans L (G) donnée par L t f(x)=f(tx). On caractérise (sous des axiomes peu restrictifs de théorie des ensembles) les fL (G) telles que l’application tL t f de G dans L (G) soit scalairement mesurable (c’est-à-dire que tϕ(L t f) est mesurable pour ϕL (G) * ). On montre que c’est le cas dès que pour tout caractère θ de L(G), tθ(L f t) est mesurable, et dans le cas compact, cela caractérise les fonctions Riemann-mesurables. On montre que l’image réciproque de tout borélien de L (G) par l’application tL 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.

@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},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {32},
     number = {1},
     year = {1982},
     pages = {39-69},
     doi = {10.5802/aif.859},
     zbl = {0452.28004},
     mrnumber = {83g:28007},
     language = {en},
     url = {http://www.numdam.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, Volume 32 (1982) no. 1, pp. 39-69. doi : 10.5802/aif.859. http://www.numdam.org/item/AIF_1982__32_1_39_0/

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