On Pettis Integrability of Translations of Functions in L~∞

摘要

Let K be a compact Hausdorff space, μ a positive Radon measure on K, and let G be a compact group with the Haar measure λ. We consider properties of the following generalization of translations on groups : we associate with every bounded μ * R~(λ-) measurable function　 f:K * G → C the function T_f:G → L~∞ (μ* R~λ) given by T_f (t) = f_t where f_t (r,s) = f(r,ts). We show that if K and G are metrizable and the class of f　 ∈ L~∞ (μ* R~λ) contains a function g such that λ({t ∈ G:(r,t) is a point of discontinuity of g}) = 0 for every r ∈ supp(μ), then T_f is Pettis integrable with respect to λ.