By John W Moon

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The Daniell-Stone type representation theorem of Greco leads us to another proof and an improvement of the Riesz type representation theorem of Sugeno, Narukawa, and Murofushi for comonotonically additive, monotone functionals. Keywords: nonadditive measure, Choquet integral, Comonotonic additivity, Greco theorem, Riesz type integral representation theorem. 1 Introduction + Let X be a locally compact Hausdorﬀ space. Let C00 (X) denote the space of all nonnegative, continuous functions on X with compact support and let C0+ (X) denote the space of all nonnegative, continuous functions on X vanishing at inﬁnity.

Two well-known nonadditive integrals are Choquet and Sugeno integrals, see [18]. The Choquet integral [2] of a measurable nonnegative function f is given by ∞ Cv (f ) = v(f t) dt 0 N ai v(Ai ) | = sup i∈I f, (Ai )N i=1 ⊂ A decreasing, ai ai 1 Ai 0, N ∈ N , i=1 where Ai+1 ⊆ Ai for every i = 1, 2, . . , N − 1, and 1Ai is the characteristic function of the set Ai . , for every A ∈ A we have ∞ 1A dv = v(A). 0 Pseudo-concave Integrals 45 The Choquet integral for ﬁnite case (and thus also for simple functions) is given on Ω = {1, .

Handbook of Measure Theory, vol. II, pp. 1403–1465. Elsevier Science, Amsterdam (2002) 21. : Maxitive measure and integration. Indag. Math. 33, 109–116 (1971) 22. : Theory of fuzzy integrals and its applications. Ph. D. Thesis. Tokyo Institute of Technology (1974) 23. : Pseudo-additive measures and integrals. J. Math. Anal. Appl. 122, 197–222 (1987) 24. : On the continuity of the concave integral. Fuzzy Sets and Systems 160, 1318–1326 (2009) 25. : Generalized Measure Theory. Springer, Boston (2009) 26.