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2007 | 5 | 4 | 619-638
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Finite-tight sets

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We introduce two notions of tightness for a set of measurable functions - the finite-tightness and the Jordan finite-tightness with the aim to extend certain compactness results (as biting lemma or Saadoune-Valadier’s theorem of stable compactness) to the unbounded case. These compactness conditions highlight their utility when we look for some alternatives to Rellich-Kondrachov theorem or relaxed lower semicontinuity of multiple integrals. Finite-tightness locates the great growths of a set of measurable mappings on a finite family of sets of small measure. In the Euclidean case, the Jordan finite-tight sets form a subclass of finite-tight sets for which the finite family of sets of small measure is composed by d-dimensional intervals. The main result affirms that each tight set H ⊆ W 1,1 for which the set of the gradients ∇H is a Jordan finite-tight set is relatively compact in measure. This result offers very good conditions to use fiber product lemma for obtaining a relaxed lower semicontinuity condition.
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Bibliografia
  • [1] J.K. Brooks and R.V. Chacon: “Continuity and compactness of measures”, Adv. in Math., Vol. 37, (1980), pp. 16–26. http://dx.doi.org/10.1016/0001-8708(80)90023-7
  • [2] Ch. Castaing and P. Raynaud de Fitte: “On the fiber product of Young measures with application to a control problem with measures”, Adv. Math. Econ., Vol. 6, (2004), pp. 1–38.
  • [3] Ch. Castaing, P. Raynaud de Fitte and A. Salvadori: “Some variational convergence results for a class of evolution inclusions of second order using Young measures”, Adv. Math. Econ., Vol. 7, (2005), pp. 1–32. http://dx.doi.org/10.1007/4-431-27233-X_1
  • [4] Ch. Castaing, P. Raynaud de Fitte and M. Valadier: Young measures on topological spaces. With applications in control theory and probability theory, Kluwer Academic Publishers, Dordrecht, 2004.
  • [5] Ch. Castaing and M. Valadier: Convex analysis and measurable multifunctions, Lecture Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin-New York, 1977.
  • [6] J. Diestel and J.J. Uhl: Vector Measures, Mathematical Surveys, no. 15, American Mathematical Society, Providence, Rhode Island, 1977.
  • [7] N. Dunford and J.T. Schwartz: Linear Operators. Part I, Reprint of the 1958 original, Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1988.
  • [8] L.C. Florescu and C. Godet-Thobie: “A Version of Biting Lemma for Unbounded Sequences in L E1 with Applications”, AIP Conference Proceedings, no. 835, (2006), pp. 58–73.
  • [9] J. Hoffmann-Jørgensen: “Convergence in law of random elements and random sets”, High dimensional probability (Oberwolfach, 1996), Progress in Probability, no. 43, Birkhäuser, Basel, 1998, pp. 151–189.
  • [10] M. Saadoune and M. Valadier: “Extraction of a good subsequence from a bounded sequence of integrable functions”, J. Convex Anal., Vol. 2, (1995), pp. 345–357.
  • [11] M. Valadier: “A course on Young measures”, Rend. Istit. Mat. Univ. Trieste, Vol. 26, (1994), suppl., pp. 349–394.
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bwmeta1.element.doi-10_2478_s11533-007-0032-2
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