Integral representation of linear functionals on spaces of unbounded functions

Let L be a vector lattice of real functions on a set Ω with 1 ∈ L, and let P be a linear positive functional on L. Conditions are given which imply the representation P (f) = ∫ fdπ, f ∈ L, for some bounded charge π. As an application, for any bounded charge π on a field F , the dual of L1(π) is shown to be isometrically isomorphic to a suitable space of bounded charges on F . In addition, it is proved that, under one more assumption on L, P is the integral with respect to a σ-additive bounded charge.