Some remarks on characterization of t-normed integrals on compacta. (English) Zbl 1543.28020
Summary: A characterization of t-normed integrals was obtained by L. M. de Campos et al. [ibid. 39, No. 1, 75–90 (1991; Zbl 0773.28015)] for finite compacta and by the author [ibid. 434, 185–197 (2022; Zbl 1522.91023)] 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 \(\sigma\)-additive regular Borel measures and the set of linear positively defined functionals. We discuss optimality of such characterization.
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