We introduce new properties of Hamel bases. We show that it is consistent with ZFC that such Hamel bases exist. Under the assumption that there exists a Hamel basis with one of these properties we construct a discontinuous and additive function that is Marczewski measurable. Moreover, we show that such a function can additionally have the intermediate value property (and even be an extendable function). Finally, we examine sums and limits of such functions.
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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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