Radon-Nikodym theorem in L-infinity

We prove that for any given set function F which satisfies F(boolean OR A(i )) = sup(i) F(A(i)) and F(A) = -infinity if meas(A) = 0, there must exist a measurable function g so that F(A) = ess sup(y is an element of A) g(y). T wo proofs of this result are given. Then a Riesz representation theorem for "linear" operators on L-infinity is proved and used to establish the exist ence of Green's function for first-order partial differential equations. In the special case u(t) + H(u, Du) = 0, Green's function is explicitly found , giving the extended Lax formula for such equations.