An approximate functional Radon-Nikodym theorem

Abstract

We prove that for two linear and positive functionals (not necessarily Daniell)J andI on a lattice unitary algebraB of functions such thatJ is absolutely continuous with respect toI, one can expressJ as follows:$J(f) = \mathop {\lim }\limits_m I(fv_m )$, where (v _m)m is a fixed sequence inB, for allf inB. This result is the “functional” similar of a previous deep result due to C. Fefferman.

The comments and the counterexamples which we are introducing show that the main result (i.e sequential approximation) cannot be improved.