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Necessary and Sufficient Conditions for the Interchange Between Infimum and the Symbol of Integration

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Abstract

We make a study of various notions of decomposability for subsets of measurable functions in relation with the interchange results between infimum and integration. For this we introduce the notions of serial decomposability and of decomposability relatively to an integrand. A characterization of closed serially decomposable subsets of the Lebesgue spaces L p is given. The second notion of decomposability introduced is characteristic for the interchange property studied. Many examples are presented. The links are made with R. T. Rockafellar’s decomposability, F. Hiai, H. Umegaki’s decomposability, G. Bouchitté and M. Valadier’s stability and normal decomposability introduced by O. Anza Hafsa and J.-P. Mandallena. As applications we obtain exact lower bounds for minimization problems of integral functionals on normally decomposable spaces (spaces of continuous functions for example), and for the minimization of a class of functionals of the Calculus of Variations.

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Correspondence to Emmanuel Giner.

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Giner, E. Necessary and Sufficient Conditions for the Interchange Between Infimum and the Symbol of Integration. Set-Valued Anal 17, 321 (2009). https://doi.org/10.1007/s11228-009-0119-y

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Keywords

  • Decomposability
  • Serial decomposability
  • Richness
  • Essential infimum
  • Continuous functions
  • Measurable integrand
  • f-decomposability
  • Integral functional
  • Calculus of Variations

Mathematics Subject Classifications (2000)

  • 26A51
  • 26B20
  • 26E15
  • 28B15
  • 28B25
  • 28B20
  • 46B42
  • 46E15
  • 46E30
  • 46N10
  • 49-XX