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Choquet-like integrals. (English) Zbl 0845.28010
The Choquet integral \(C(f, m)\) of a non-negative measurable function \(f\) with respect to a real-valued monotone set function \(m\) with \(m(\emptyset)= 0\) is defined by the equality \(C(f, m)= \int_{[0, \infty]} m(f\geq x) dx\). In the paper Choquet-like integrals are studied working with so-called pseudoaddition \(\oplus\) and pseudomultiplication \(\odot\) instead of usual addition \(+\) and multiplication \(\cdot\). A special attention is given to the case when \(a\oplus b= \max(a, b)\) and to the case, when \(a\oplus b= g^{- 1}(g(a)+ g(b))\), where \(g\) is a convenient generator. The author compares the Choquet construction with another one given by M. Sugeno and T. Murofushi [J. Math. Anal. Appl. 122, 197-222 (1987; Zbl 0611.28010)].

28E10 Fuzzy measure theory
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