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Monotone set functions-based integrals. (English) Zbl 1099.28007
Pap, E. (ed.), Handbook of measure theory. Vol. I and II. Amsterdam: North-Holland (ISBN 0-444-50263-7/hbk). 1329-1379 (2002).
This paper is a chapter of Pap’s handbook on measure theory and presents an excellent review of integral theories of the Choquet type. The main advantage of these theories is a possibility to apply them also for the case of non-additive measures. This fact is very important from the point of view of some applications, e.g., the modelling of the weights of criterion groups in multi-criteria decision making.
Nearby the definition of the Choquet integral is genially simple. If \(\mu:{\mathcal A}\to [0,\infty]\) is a non-decreasing function defined on a \(\sigma\)-algebra \({\mathcal A}\) of subsets of \(\Omega\) with \(\mu(\emptyset) \,\overline{.\,.}\,0\), and \(f\) is a non-negative \({\mathcal A}\)-measurable function, then the Choquet integral can be defined by the formula \[ \int f\,d\mu= \int_0^\infty \mu(\{\omega\in\Omega;\;f(\omega)>t\})\,d\omega.\tag{C} \] A similar definition has the Sugeno integral: \[ \int f\,d\mu\,\overline{.\,.}\, \bigvee_{t\in[0,1]} t\wedge \mu(\{\omega\in\Omega;\;f(\omega)>t\}).\tag{S} \] Therefore the authors give the main attention to a general theory containing both theories (Choquet and Sugeno) as special cases. The general theory is constructed with respect to axiomatically given operations \(\oplus, \odot\). If \(\oplus, \odot\) are the usual sum and product of real numbers, resp., then one obtains the Choquet integral; if these operations are \(\vee,\wedge\), then one obtains the Sugeno integral.
The chapter consists of five sections. After Introduction the authors recall, discuss and compare in Section 2 the properties of the Choquet integral and the Sugeno integral. Section 3 is devoted to an appropriate choice of the binary operations \(\oplus\) and \(\odot\). The construction of the general fuzzy integral is done in Section 4. Finally, in Section 5 there are given several examples of operations \(\oplus\) and \(\odot\) with the corresponding fuzzy integrals.
The general fuzzy integral is constructed in detail. For the integral \(\int^\oplus f\odot d\mu\) defined on a family of non-negative \({\mathcal A}\)-measurable functions the authors request axiomatically four properties: coincidence with expected values in the case of a constant function on a set, monotonicity, comonotone \(\oplus\)-additivity, and horizontal \(\oplus\)-additivity. Moreover, if \(\mu\) is continuous from below, then the continuity of the integral from below is requested, too. These properties lead to the formula \(\int^\oplus f\odot d\mu=\sup\{L^\oplus(s)\); \(s\leq f\), \(s\) is simple}, where \(L^\oplus(s)\) is an appropriate integral of the simple function \(s\). The integral satisfies all the requested properties, and also some further ones.
Also the case of the whose real line as a range of the integrated function \(f\) is considered. Here there are two possibilities: symmetric theory, and asymmetric theory. Symmetric case was first studied by Šipoš \[ \int\,d\mu=\int f^+-\int f^-\, d\mu^d \tag{\v S} \] where \(\mu^d(A)= \mu(\Omega)- \mu(A^c)\). Let me mention the following 25 years old story. J. Šipoš submitted his three papers about his general integral to the redaction of the journal Mathematica Slovaca. One of the paper was recommended to be refused by the referee. Of course the members of the editorial board L. Mišík and T. Neubrunn saved Šipoš’ excellent work suggesting to send the manuscript to another referee.
Examples present the strongest property of the chapter. Let us mention Pap’s \(g\)-integral, the integrals of Sugeno, Ralescu-Adams, Weber as well as many new constructions. A very encouraging list of references is included. Listed papers and books contain many new results. The reader can obtain good orientation in the theory as well as a hopeful look for applications.
For the entire collection see [Zbl 0998.28001].

28E10 Fuzzy measure theory
28A25 Integration with respect to measures and other set functions
28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections