Finite-tight sets

• Autorzy: Liviu Florescu
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

finite-tight set; Jordan finite-tight set; Young measure; w2 - convergence

Kategorie tematyczne

• 28A20: Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
• 49J45: Methods involving semicontinuity and convergence; relaxation
• 46E27: Spaces of measures
• 28A33: Spaces of measures, convergence of measures

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2007
• Tom: 5
• Numer: 4
• Strony: 619-638

Daty

wydano

2007-12-01

online

2007-12-01

Twórcy

autor

Liviu Florescu

• “Al. I. Cuza” University

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.

Identyfikatory

DOI

10.2478/s11533-007-0032-2