Decomposition of positive definite functions defined on a neighbourhood of the identity

LetV be a symmetric open neighbourhood of the identity of a topological groupG. We show that every positive definite functionf onV can be written asf=f _c +f _s wheref _c andf _s are positive definite functions onV, f _c is continuous andf _s averages to zero. IfG is locally compact with Haar measurem _G andf ism _G -measurable thenf _s =0m _G -almost everywhere.