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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)].

##### MSC:
 2.8e+11 Fuzzy measure theory
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