ON THE MEASURABILITY OF ADDITIVE FUNCTIONALS

Abstract. For an infinite-dimensional separable Hilbert space H, the prob-lem of measurability of additive functionals f: H → R with respect to various extensions of σ-finite diffused Borel measures on H is discussed. It is shown that there exists an everywhere discontinuous additive functional f on H such that, for any σ-finite diffused Borel measure µ on H, this f can be made measurable with respect to an appropriate extension of µ. Special consideration is given to the case where µ is invariant or quasiinvariant under a subgroup of H