Abstract.
We prove that for any given set function Fwhich satisfies F(∪ A i ) =supi F(A i )and F(A)=-∈ftyif meas (A)=0 , there must exist a measurable function gso that F(A)= ess sup_y ∈Ag(y) . Two proofs of this result are given. Then a Riesz representation theorem for ``linear'' operators on L ∈ftyis 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.
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Accepted 20 March 2000. Online publication 7 July 2000.
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Barron, E., Cardaliaguet, P. & Jensen, R. Radon—Nikodym Theorem in L∞. Appl Math Optim 42, 103–126 (2000). https://doi.org/10.1007/s002450010006
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DOI: https://doi.org/10.1007/s002450010006