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Transactions of the American Mathematical Society

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ISSN 1088-6850 (online) ISSN 0002-9947 (print)

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Densities of idempotent measures and large deviations
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by Marianne Akian PDF
Trans. Amer. Math. Soc. 351 (1999), 4515-4543 Request permission

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
  • Email: marianne.akian@inria.fr
  • Received by editor(s): June 16, 1995
  • Received by editor(s) in revised form: April 17, 1997
  • Published electronically: July 19, 1999
  • © Copyright 1999 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 351 (1999), 4515-4543
  • MSC (1991): Primary 28B15, 49J52; Secondary 06B35, 60F10
  • DOI: https://doi.org/10.1090/S0002-9947-99-02153-4
  • MathSciNet review: 1466943