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

Michel Talagrand (1982)

Annales de l'institut Fourier

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

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