The Haar theorem for lattice-ordered abelian groups with order-unit

Given a finite union P of rational Simplexes, we assign to P numerical invariants λ0, λ1;..., λ dim P; each λi is the suitably normalized volume of the i-dimensional part of P. We then prove that every finitely generated projective lattice-ordered abelian group G with order-unit u has a faithful invariant positive linear functional s : (7 → K. For each g £ G, s(g) is the integral of g over the maximal spectrum of G, the latter being canonically identified with a rational polyhedron P. Volume elements are measured by the λi's. The proof uses the polyhedral versions of the Wlodarczyk-Morelli theorem on decompositions of birational toric maps in blow-ups and blow-downs, and of the De Concini-Procesi theorem on elimination of points of indeterminacy.