Most applications of non-additive measures in game theory, decision theory, economics, and related fields do not focus on measures as set functions, but rather on integrals that allow the computation of expected utility or expected payoff. Several types of integrals with respect to non-additive measures have been developed. These integrals are commonly referred to as fuzzy integrals.

One of the central problems in the theory of fuzzy integrals is to characterize such integrals as functionals on suitable function spaces (see, for example, Subchapter 4.8 in [Grabisch, 2016], which is devoted to characterizations of the Choquet and Sugeno integrals).
We can consider these results as non-additive analogues of the well-known Riesz representation theorem establishing a correspondence between the set of $\sigma$-additive regular Borel measures and the set of positive linear functionals. A characterization of t-normed integrals was obtained in [de Campos, Lamata and Moral, 1991] for finite compacta, and extended to the general case in [Radul, 2022].

It was shown in [Grabisch, 2016] that in the special case of the Sugeno integral over finite sets, some of the conditions used in the characterization theorem of [de Campos, Lamata and Moral, 1991] - notably monotonicity - are redundant, and a simpler characterization was proposed. This result was later generalized to the case of t-normed integrals on compacta in [Radul, 2023]. In particular, it was shown that monotonicity follows from other properties of the t-normed integral, among them the comonotonic maxitivity property.

The question of whether comonotonic maxitivity alone implies monotonicity was explicitly posed in [Radul, 2023], where a positive answer was established for finite compacta. We show that this implication does not hold in general and formulate several related open problems.

\noindent\textbf{References}

\noindent L. M. de Campos, M. T. Lamata and S. Moral (1991) {\em A unified approach to define fuzzy integrals}, Fuzzy Sets and Systems 39, 75--90.

\noindent M. Grabisch (2016), Set Functions, Games and Capacities in Decision Making. Springer.

\noindent T. Radul (2022), {\em Games in possibility capacities with payoff expressed by fuzzy integral}, Fuzzy Sets and systems 434, 185-197.

\noindent T. Radul (2023), {\em Some remarks on characterization of t-normed integrals on compacta}, Fuzzy Sets and systems 467, 108490.