Titlebook: Non-Additive Measure and Integral; Dieter Denneberg Book 1994 Springer Science+Business Media Dordrecht 1994 artificial intelligence.bound

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Representing Functionals as Integrals,eory) is under what conditions the representing set function is sub- or supermodular and continuous from below. A corollary of the respective Representation Theorem is the classical Daniell-Stone Representation Theorem, where the representing set function is a measure.

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Integration of Monotone Functions on Intervals,urvey integration of monotone functions. We are working with countable subdivisions to include the improper Riemann integral from the beginning. Crucial for later chapters will be the pseudo-inverse function of a decreasing function. It is introduced in the present chapter.

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The Asymmetric Integral,aves asymmetric. In Chapter 7 we shall modify the definition in order to get a symmetric and fully homogenous integral. An important property of the asymmetric integral, not shared by the symmetric one, is comonotonic additivity.

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The Symmetric Integral,tions our old integral and the new one differ in two relevant points: asymmetry is replaced by symmetry and comonotonic additivity is lost for functions essentially assuming positive and negative values.

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Families of Measures and their Envelopes,r the integrals in the family. The main result is a characterization of submodular set functions by means of envelopes of additive set functions. The method of generating a set function as supremum of a given family of set functions will be employed, too, for proving the Radon-Nikodym Theorem in the next chapter.

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