Niveloids
The authors introduce a class of functionals named niveloids, defined on the set of all extended real-valued functions on a set \(X\), as follows: \[ {\mathcal N}(X) = \{T: \overline{\mathbb{R}}^X\mapsto \overline{\mathbb{R}};\;T\text{ is isotone and vertically invariant}\}, \] where the both defining notions mean, respectively, the properties: \[ f\leq g\Rightarrow Tf\leq Tg,\tag{1} \] \[ \forall r\in \mathbb{R}\;\;T(f+ r)= T(f)+ r.\tag{2} \] Among the niveloids one can find, for instance, limitoids of G. H. Grecco, \(\Gamma\)-limits of E. De Giorgi, \(Q\)-limits of L. Cesari, etc. Some operations (like upper and lower projections) on functionals as elements of the complete lattice \({\mathcal F}(X)\) of all the functionals ordered pointwise are defined and some representation theorems for niveloids are given. Other representation theorems for niveloids are obtained by using the inf- and sup-convolutions on functions from \(\overline{\mathbb{R}}^X\). Besides, some uniform and improper topologies on functions from \(\overline{\mathbb{R}}^X\) are introduced so that niveloids would be upper (resp. lower) semicontinuous. Also, several other notions are introduced and various links between them are established as, for instance, a lattice structure for the set of niveloids is considered and an isomorphism with the complete lattice of closed sets with respect to the uniform topology is proved.
scientific article; zbMATH DE number 807911