A Radon-Nikodym theorem for finitely additive set functions

Abstract

Suppose that is a field of subsets of the set S, and suppose that and are complex-valued finitely additive set functions defined on . Assume that is bounded and is finite and absolutely continuous with respect to . (A word of warning is in order here. The statement " is absolutely continuous with respect to " is often interpreted as "(E) = 0 implies (E) = 0". This is not the meaning used here. Our definition is "for every e > 0 there is a > 0 such that | (E) \ < implies I (E) I < ." Unless is bounded and countably additive, the two definitions are not equivalent.) THEOREM 1. There exists a sequence {f n } of /^-simple functions on S, such that