# Radon—Nikodym Theorem in L ∞

Radon—Nikodym Theorem in L ∞ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

# Radon—Nikodym Theorem in L ∞

, Volume 42 (2) – Jan 1, 2000
24 pages

Publisher
Springer-Verlag
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics, Simulation
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s002450010006
Publisher site
See Article on Publisher Site

### 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.

### Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Jan 1, 2000

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