Null-additive set functions
This volume presents a unified approach to the mathematical theory of a wide class of non-additive set functions, the so called null-additive set functions, which also includes classical measure theory. It includes such important set functions as capacities, triangular set functions, some fuzzy measures, submeasures, decomposable measures, possibility measures, distorted probabilities, autocontinuous set functions, etc. The usefulness of the theory is demonstrated by applications in nonlinear differential and difference equations; fractal geometry in the theory of chaos; the approximation of functions in modular spaces by nonlinear singular integral operators; and in the theory of diagonal theorems as a universal method for proving general and fundamental theorems in functional analysis and measure theory. Audience: This book will be of value to researchers and postgraduate students in mathematics, as well as in such diverse fields as knowledge engineering, artificial intelligence, game theory, statistics, economics, sociology and industry.
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Additive Representations of Set Function
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A G E a-algebra a-ring absolutely continuous additive set functions arbitrary atom B C A bounded box-counting dimension Choquet integral condition continuous set function continuous with respect convergence Corollary D-poset decomposable measures Definition denote diagonal theorems disjoint sequence disjoint sets disjoint variation element Example f dm finite fndm Hence idempotent implies inequality Lebesgue Lebesgue integral Lemma m(En measurable function measure space measure theory monotone measure space monotone set function negative set neighbourhood system non-additive set functions non-negative null-additive set functions obtain open sets order continuous orthomodular lattice positive monotone set positive set Proof Propositon prove pseudo-addition real numbers representation s-null set satisfies semigroup set function defined SF(f signed fuzzy measure signed measure simple function subadditive submeasure subset superadditive supermodular Suppose topological topological ring triangular set functions type theorem