Integral representation of linear functionals on spaces of unbounded functions

Let L be a vector lattice of real functions on a set Omega with 1 is an element of L, and let P be a linear positive functional on L. Conditions are given which imply the representation P(f) = integral fd pi, f is an element of L, for some bounded charge pi. As an application, for any bounded charge pi on a field F, the dual of L-1(pi) 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 sigma-additive bounded charge.