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Densities of idempotent measures and large deviations

Author: Marianne Akian
Journal: Trans. Amer. Math. Soc. 351 (1999), 4515-4543
MSC (1991): Primary 28B15, 49J52; Secondary 06B35, 60F10
Published electronically: July 19, 1999
MathSciNet review: 1466943
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Abstract: Considering measure theory in which the semifield of positive real numbers is replaced by an idempotent semiring leads to the notion of idempotent measure introduced by Maslov. Then, idempotent measures or integrals with density correspond to supremums of functions for the partial order relation induced by the idempotent structure. In this paper, we give conditions under which an idempotent measure has a density and show by many examples that they are often satisfied. These conditions depend on the lattice structure of the semiring and on the Boolean algebra in which the measure is defined. As an application, we obtain a necessary and sufficient condition for a family of probabilities to satisfy the large deviation principle.

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Additional Information

Marianne Akian
Affiliation: INRIA, Domaine de Voluceau, B.P. 105, 78153 Le Chesnay Cedex, France

Keywords: Idempotent semiring, dioid, max-plus algebra, continuous lattice, idempotent measure, optimization, large deviations.
Received by editor(s): June 16, 1995
Received by editor(s) in revised form: April 17, 1997
Published electronically: July 19, 1999
Article copyright: © Copyright 1999 American Mathematical Society