By Michio Sugeno (auth.), Shoumei Li, Xia Wang, Yoshiaki Okazaki, Jun Kawabe, Toshiaki Murofushi, Li Guan (eds.)

This quantity is a suite of papers awarded on the overseas convention on Nonlinear arithmetic for Uncertainty and Its functions (NLMUA2011), held at Beijing collage of know-how throughout the week of September 7--9, 2011. The convention introduced jointly major researchers and practitioners concerned with all features of nonlinear arithmetic for uncertainty and its applications.

Over the final fifty years there were many makes an attempt in extending the idea of classical chance and statistical types to the generalized one that can deal with difficulties of inference and selection making while the model-related info is scarce, obscure, ambiguous, or incomplete. Such makes an attempt contain the examine of nonadditive measures and their integrals, obscure chances and random units, and their purposes in details sciences, economics, finance, coverage, engineering, and social sciences.

The booklet offers subject matters together with nonadditive measures and nonlinear integrals, Choquet, Sugeno and different sorts of integrals, chance concept, Dempster-Shafer thought, random units, fuzzy random units and comparable records, set-valued and fuzzy stochastic procedures, obscure likelihood concept and comparable statistical versions, fuzzy arithmetic, nonlinear practical research, details concept, mathematical finance and threat managements, choice making lower than numerous sorts of uncertainty, and others.

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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.