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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...

Closed ideals in certain Beurling algebras, and synthesis of hyperdistributions

J. Esterle (1996)

Studia Mathematica

Closed Ideals in Subalgebras of Banach Algebras II: Ditkin's Condition.

J.T. Burnham (1974)

Monatshefte für Mathematik

Closed Ideals of Homogeneous Algebras.

J.A. Ward (1983)

Monatshefte für Mathematik

Closed subgroups of nuclear spaces are weakly closed

Wojciech Banaszczyk (1984)

Studia Mathematica

Coarea integration in metric spaces

Malý, Jan (2003)

Nonlinear Analysis, Function Spaces and Applications

Let $X$ be a metric space with a doubling measure, $Y$ be a boundedly compact metric space and $u : X \to Y$ be a Lebesgue precise mapping whose upper gradient $g$ belongs to the Lorentz space $L_{m, 1}$ , $m \geq 1$ . Let $E \subset X$ be a set of measure zero. Then ${\hat{ℋ}}_{m} (E \cap u^{- 1} (y)) = 0$ for $ℋ_{m}$ -a.e. $y \in Y$ , where $ℋ_{m}$ is the $m$ -dimensional Hausdorff measure and ${\hat{ℋ}}_{m}$ is the $m$ -codimensional Hausdorff measure. This property is closely related to the coarea formula and implies a version of the Eilenberg inequality. The result relies on estimates of Hausdorff content of level sets...