AbstractL-spaces and the Radon-Nikodym theorem

Abstract

Let (X, A, μ) be a measure space and letL=L 1(X). Then everyE∈A can be considered as a projector onL via Ef =f ⋅ χE. We seek to replaceL by an arbitraryAL-space andA by a collectionR of operators onL with as little structure as is necessary to guarantee that “absolutely continuous set functions have integral representations”. By “set function” we mean an additive functional onR and by “integral” we mean the natural abstract integralm(f)=‖f +‖-‖f−‖ defined on anyAL-space. We takeR to be what we call aB-ring of contractors, a generalization of Boolean ring of sets (projectors). This is interesting sinceR need not be a lattice, thus in a sense the fact that the measurable sets have a Boolean structure is inessential. But we must postulate a Riesz Decomposition Property forR. So thatR is “large enough”, we assume thatR separates L in the sense of S. Leader. A theory is developed which culminates in the followingRadons-Nikodym Theorem: LetR be a separatingB-ring of contractors on theAL-spaceL. Suppose that ϕ is a bounded additive functional onR such that if {E α} is a net of elements ofR withE α f→0 for everyf∈L, then necessarily ϕ(E α)→0. Then there existsf∈L such that ϕ(E)=m(Ef) for everyE∈R.