Appl Math Optim 42:103–126 (2000)
2000 Springer-Verlag New York Inc.
Radon–Nikodym Theorem in L
E. N. Barron,
and R. R. Jensen
Department of Mathematical & Computer Sciences,
Loyola University Chicago,
Chicago, IL 60626, USA
Centre de Recherche Viabilit´e, Jeux, Controle,
Universit´e de Paris Dauphine,
Paris 75775 Cedex, France
Abstract. We prove that for any given set function F which satisﬁes F(
) and F(A) =−∞if meas(A) = 0, there must exist a measurable
function g so that F(A) = ess sup
g(y). Two proofs of this result are given.
Then a Riesz representation theorem for “linear” operators on L
is proved and
used to establish the existence of Green’s function for ﬁrst-order partial differential
equations. In the special case u
+ H (u, Du) = 0, Green’s function is explicitly
found, giving the extended Lax formula for such equations.
Key Words. Radon–Nikodym, Riesz, Lax formula.
AMS Classiﬁcation. 28A10, 28A25, 35C.
The Radon–Nikodym theorem for absolutely continuous measures is a cornerstone of the
theory of integration. It is a fundamental result with applications throughout analysis.
Within analysis one of the main applications is the Riesz representation theorem for
E. N. Barron and R. R. Jensen were supported in part by Grants DMS-9532030 and DMS-9972043
from the National Science Foundation.