Finite-tight sets

Liviu Florescu

Open Mathematics (2007)

  • Volume: 5, Issue: 4, page 619-638
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Liviu Florescu. "Finite-tight sets." Open Mathematics 5.4 (2007): 619-638. <http://eudml.org/doc/269450>.

@article{LiviuFlorescu2007,
abstract = {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.},
author = {Liviu Florescu},
journal = {Open Mathematics},
keywords = {finite-tight set; Jordan finite-tight set; Young measure; w2 - convergence},
language = {eng},
number = {4},
pages = {619-638},
title = {Finite-tight sets},
url = {http://eudml.org/doc/269450},
volume = {5},
year = {2007},
}

TY - JOUR
AU - Liviu Florescu
TI - Finite-tight sets
JO - Open Mathematics
PY - 2007
VL - 5
IS - 4
SP - 619
EP - 638
AB - 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.
LA - eng
KW - finite-tight set; Jordan finite-tight set; Young measure; w2 - convergence
UR - http://eudml.org/doc/269450
ER -

References

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  3. [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 Zbl1145.49005
  4. [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. [5] Ch. Castaing and M. Valadier: Convex analysis and measurable multifunctions, Lecture Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin-New York, 1977. 
  6. [6] J. Diestel and J.J. Uhl: Vector Measures, Mathematical Surveys, no. 15, American Mathematical Society, Providence, Rhode Island, 1977. Zbl0369.46039
  7. [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. [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. [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. [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. Zbl0833.46018
  11. [11] M. Valadier: “A course on Young measures”, Rend. Istit. Mat. Univ. Trieste, Vol. 26, (1994), suppl., pp. 349–394. Zbl0880.49013

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