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

Compactness and Tightness in a Space of Measures with the Topology of Weak Convergence.

Flemming Topsoe (1974)

Mathematica Scandinavica

Compactness in the sense of the convergence with respect to a small system

Jacek Hejduk, Eliza Wajch (1989)

Mathematica Slovaca

Compacts de fonctions de première classe

Michèle Capon (1975/1976)

Séminaire Choquet. Initiation à l'analyse

Completeness of the set of sub- $σ$ -algebras.

Wang, Xikui (1993)

International Journal of Mathematics and Mathematical Sciences

Continuous-, derivative-, and differentiable-restrictions of measurable functions

Jack Brown (1992)

Fundamenta Mathematicae

We review the known facts and establish some new results concerning continuous-restrictions, derivative-restrictions, and differentiable-restrictions of Lebesgue measurable, universally measurable, and Marczewski measurable functions, as well as functions which have the Baire properties in the wide and restricted senses. We also discuss some known examples and present a number of new examples to show that the theorems are sharp.

Contre-exemples associés aux théorèmes de Rosenthal. Quelques propriétés liées à ces résultats

Gilles Godefroy (1975/1976)

Séminaire Choquet. Initiation à l'analyse

Convergence almost everywhere of sequences of measurable functions

Elżbieta Wagner, Władysław Wilczyński (1981)

Colloquium Mathematicae

Convergence diffuse d'une suite de fonctions

Jean Raymond (1981)

Fundamenta Mathematicae

Convergence of conditional expectations

Josef Štěpán (1977)

Commentationes Mathematicae Universitatis Carolinae

Convergence of sequences of real functions with respect to small systems

Jerzy Niewiarowski (1988)

Mathematica Slovaca

Convergence theorems for a Daniell-Loomis integral.

Günzler, H. (1991)

Mathematica Pannonica

Convergence theorems for the PU-integral

Giuseppa Riccobono (2000)

Mathematica Bohemica

We give a definition of uniform PU-integrability for a sequence of $μ$ -measurable real functions defined on an abstract metric space and prove that it is not equivalent to the uniform $μ$ -integrability.

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