⊥-Decomposable measures and integrals for Archimedean t-conorms ⊥

A characterization of t-normed integrals was obtained by Luis M. de Campos, María T. Lamata and Serafín Moral (1991) for finite compacta and by T. Radul (2022) for the general case. Such characterization establishes a correspondence between the space of capacities and homogeneous respect t-norm monotone normalized functionals preserving the maximum operation of comonotone functions. In fact, we can consider these theorems as non-additive and non-linear analogues of well-known Riesz Theorem about a correspondence between the set of σ-additive regular Borel measures and the set of linear positively defined functionals. We discuss optimality of such characterization.