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.
@article{bwmeta1.element.doi-10_2478_s11533-007-0032-2,
author = {Liviu Florescu},
title = {Finite-tight sets},
journal = {Open Mathematics},
volume = {5},
year = {2007},
pages = {619-638},
zbl = {1155.28301},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-007-0032-2}
}
Liviu Florescu. Finite-tight sets. Open Mathematics, Tome 5 (2007) pp. 619-638. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-007-0032-2/
[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 | Zbl 0463.28003
[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. | Zbl 1137.28300
[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 | Zbl 1145.49005
[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. | Zbl 0369.46039
[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. | Zbl 0833.46018
[11] M. Valadier: “A course on Young measures”, Rend. Istit. Mat. Univ. Trieste, Vol. 26, (1994), suppl., pp. 349–394. | Zbl 0880.49013