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Equiregulated Sets

      In this subsection, our goal is to investigate important properties of equiregulated sets. In addition to [97], the reference [40] also deals with a characterization of subsets of equiregulated functions.

      

      Definition 1.10: A set is called equiregulated, if it has the following property: for every and every , there exists such that

      1 if , and , then ;

      2 if , and , then .

      The next result, which can be found in [97, Proposition 3.2], gives a characterization of equiregulated functions taking values in a Banach space.

      Theorem 1.11: A set is equiregulated if and only if for every , there is a division of such that

(1.1)

       for every and , for .

Proof. Let be given and let be the set of all such that there is a division , that is, for which (1.1) holds with instead of . Since is equiregulated, there is such that for every and . Let and . Thus, for and , we have

      Hence, .

      Let be the supremum of the set . Since , is regulated. Thus, there exists such that for every and . Take and a division , say, , such that (1.1) holds with instead of . Denote . Then, for and , we have

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