Extension of a Theorem of Bochner on Express- Ing Functionals as Riemann Integrals

Introduction. S. Bochner has shown that an additive homogeneous functional defined over a sufficiently large class C of functions can be realized as a Riemann integral with respect to a finitely additive measure V in the space X over which the functions are defined. His proof makes use of the fact that the constant function belongs to C, as a result, V(X) is finite. I t is the purpose of this note to show that a similar theorem holds even when V(X) turns out to be infinite. A modification of Bochner's proof would suffice for this stronger theorem. We have chosen rather to treat it as a problem of extending the domain of definition of the given functional. Throughout we have used the symbol —> to be read as "implies." The equality ss is used to denote an equality which holds by definition.