Conditional expectations for general measure spaces

Abstract

The classical conditional expectation with respect to σ-algebras, on probability measure spaces, has been extended for infinite measure spaces. In this paper we consider conditional expectations with respect to δ-rings, on arbitrary measure spaces. An additional condition has to be imposed in order to insure the uniqueness of the conditional expectation. The existence of the conditional expectation is proved for functions in L p with 1 ≤ p < ∞, and, for localizable measures, also in L ∞. The properties of the classical conditional expectation remain true in the general case, sometimes with some modifications.