Radon—Nikodym Theorem in L∞

Abstract

We prove that for any given set function F which satisfies F(∪ A i ) =sup i F(A i ) and F(A)=-∈fty if meas (A)=0 , there must exist a measurable function g so that F(A) = ess sup_ y ∈ A g(y) . Two proofs of this result are given. Then a Riesz representation theorem for ``linear'' operators on L ∈fty is proved and used to establish the existence 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.