Integral representation of linear functionals on spaces of unbounded functions

Abstract

Let L L be a vector lattice of real functions on a set Ω \Omega with 1 ∈ L \boldsymbol {1}\in L , and let P P be a linear positive functional on L L . Conditions are given which imply the representation P ( f ) = ∫ f d π P(f)=\int fd\pi , f ∈ L f\in L , for some bounded charge π \pi . As an application, for any bounded charge π \pi on a field F \mathcal F , the dual of L 1 ( π ) L^1(\pi ) is shown to be isometrically isomorphic to a suitable space of bounded charges on F \mathcal F . In addition, it is proved that, under one more assumption on L L , P P is the integral with respect to a σ \sigma -additive bounded charge.