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Finite-tight sets. (English) Zbl 1155.28301

Summary: 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 \(H\subseteq W^{1,1}\) for which the set of the gradients \(\nabla H\) 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.

MSC:

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