Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Bücher
Zeitschriften
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
ATZ-Fachkongress

Chassis.tech plus 2024


4 Kongresse in einer Veranstaltung

Treffen Sie in München die Fachleute von Chassis, Lenkung, Bremsen sowie Reifen/Räder.
Erweiterte Suche
Anmelden
nach oben

1994 | Buch

Integration of Monotone Functions on Intervals Kapitel lesen Erstes Kapitel lesen

Non-Additive Measure and Integral

verfasst von: Dieter Denneberg

Verlag: Springer Netherlands

Buchreihe : Theory and Decision Library

Enthalten in: Professional Book Archive

Einloggen, um Zugang zu erhalten
insite
SUCHEN

Über dieses Buch

Non-Additive Measure and Integral is the first systematic approach to the subject. Much of the additive theory (convergence theorems, Lebesgue spaces, representation theorems) is generalized, at least for submodular measures which are characterized by having a subadditive integral. The theory is of interest for applications to economic decision theory (decisions under risk and uncertainty), to statistics (including belief functions, fuzzy measures) to cooperative game theory, artificial intelligence, insurance, etc.
Non-Additive Measure and Integral collects the results of scattered and often isolated approaches to non-additive measures and their integrals which originate in pure mathematics, potential theory, statistics, game theory, economic decision theory and other fields of application. It unifies, simplifies and generalizes known results and supplements the theory with new results, thus providing a sound basis for applications and further research in this growing field of increasing interest. It also contains fundamental results of sigma-additive and finitely additive measure and integration theory and sheds new light on additive theory. Non-Additive Measure and Integral employs distribution functions and quantile functions as basis tools, thus remaining close to the familiar language of probability theory.
In addition to serving as an important reference, the book can be used as a mathematics textbook for graduate courses or seminars, containing many exercises to support or supplement the text.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Integration of Monotone Functions on Intervals
Abstract
Our approach to the general theory of integration is based, via distribu­tion functions, upon the integral of monotone functions on intervals. The latter is already provided by the (improper) Riemann integral. For the sake of completeness, to fix the terminology and to prepare subsequent proofs we survey 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.
Dieter Denneberg
Chapter 2. Set Functions and Caratheodory Measurability
Abstract
Here set functions and their elementary properties are introduced and the classical Caratheodory extension process for σ-additive set functions is performed with a slightly new view: We separate additivity from continuity assumptions on the set function and emphasize submodularity, a weaker property than additivity. In this section we mainly rearrange well known proofs or methods of proof which we found mainly in Ash 1972, Kelley-Srinivasan 1988 and M.M.Rao 1987.
Dieter Denneberg
Chapter 3. Construction of Measures using Topology
Abstract
If one wants to show that usual length on the real line is σ-additive one needs that IR is locally compact. More generally, a topology on Ω with sufficiently many compact sets allows to construct broad classes of measures starting with finitely additive set functions which are regular, i.e. compatible with the given topology. On the real line, essentially these are the Lebesgue-Stieltjes measures. The main idea of this construction generalizes to establish continuity from below for a subadditive, regular set function (Exercise 3.5).
Dieter Denneberg
Chapter 4. Distribution Functions, Measurability and Comonotonicity of Functions
Abstract
For a function we introduce the system of upper level sets. Together with a set function it gives rise to the decreasing distribution function and, in the next chapter, the integral. No measurability requirements have to be imposed on the function if the set function is defined on the whole power set. For many questions this can be supposed but for some topics (Radon-Nikodym-Theorem, conditional expectation) set functions with restricted domains are crucial. In this situation a function is called measurable if a unique distribution function can be assigned to it. Greco’s characterization of measurability will be given and we prove the important theorem on measurability of sums of measurable functions.
Dieter Denneberg
Chapter 5. The Asymmetric Integral
Abstract
By means of the quantile function and her integral in the sense of Chapter 1 we define the integral of a function with respect to an arbitrary monotone set function μ. This integral cannot be additive if μ is not and it is only positively homogenous. With respect to multiplying by-1 the integral behaves 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.
Dieter Denneberg
Chapter 6. The Subadditivity Theorem
Abstract
If the integral with respect to a monotone set function µ is subadditive, i.e.
$$\int {(X + Y)d\mu \int {Xd\mu + \int {Yd\mu } } } $$
then µ is submodular (cf. Exercise 5.1). Here we shall prove that submodularity of the set function is also sufficient for subadditivity of the integral. The corresponding theorems for supermodular and additive set functions are corollaries.
Dieter Denneberg
Chapter 7. The Symmetric Integral
Abstract
For additive, finite set functions the integral introduced in Chapter 5 is the usual one. For infinite measures this is true only for nonnegative functions. The alternative integral to be defined here, coincides with the usual integral in all cases of additive set functions. For nonadditive set functions 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.
Dieter Denneberg
Chapter 8. Sequences of Functions and Convergence Theorems
Abstract
The convergence theorems we start with yield sufficient conditions that the integral can be interchanged with pointwise convergence of functions. These theorems clearly require continuity of the set function. First the Monotone Convergence Theorem derives from the special case proved in Chapter 1. It is valid very generally for monotone set functions. The other important theorem, Lebesgue’s Dominated Convergence Theorem, is first proved for the classical case of measures and later on it is generalized for subadditive set functions in applying the classical case with Lebesgue measure on the distribution functions. For this purpose we weaken pointwise convergence to stochastic convergence and further to convergence in distribution. These different types of convergence are important, too, in probability theory and statistics. By the way we get some information on measurability of the limit function even if the set function is not continuous.
Dieter Denneberg
Chapter 9. Nullfunctions and the Lebesgue Spaces Lp
Abstract
The integral with respect to a submodular set function allows to define norms on spaces of measurable functions. An obstacle in doing so is that (like in the σ-additive theory) there are functions, not identically zero, having norm nought. Those are the nullfunctions, we start with. Then the Lebesgue space L 1(µ) of a submodular µ is shown to be a normed linear space and to be a Banach space if µ is continuous from below. These results translate to L ∞(µ) via the identity L ∞(µ) = L 1(sign µ). The spaces L p , 1 < p < ∞, are defined but not treated in detail. At the end of the chapter we show that assigning the quantile function to a function is a continuous even contracting operator χ. Recall that this assignment was the first step in Choquet’s and our approach to integration theory which now turns out to have good topological and metric properties. Furthermore χ is piecewise linear, namely on cones of comonotonic functions.
Dieter Denneberg
Chapter 10. Families of Measures and their Envelopes
Abstract
Families of measures play important roles in statistics and economic decision theory in order to model uncertainty. They also appear as the core of a cooperative game in game theory. The envelopes (supremum or infimum) of the members of the family are set functions and their integrals give bounds for 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.
Dieter Denneberg
Chapter 11. Densities and the Radon-Nikodym Theorem
Abstract
If a set function µ on an algebra A ⊂ 2Ω is given one can modify µ to a new set function v on A by means of a so called density function on Ω. Such v is absolutely continuous with respect to µ, v ≪ µ. In case of measures the condition v ≪ µ is also sufficient for v having a density. This is the important Radon-Nikodym theorem. Closely related to these questions is the problem of representing a given functional on a function space through an integral. Two such theorems, one for the non-additive case, the other, well known, for the additive case are given here. More general representation theorems will be given in Chapter 13.
Dieter Denneberg
Chapter 12. Products
Abstract
In probability theory, the generalized product of measures is a constructive way for introducing conditional expectation. The abstract definition of conditional expectation essentially uses the Radon-Nikodym Theorem, so it works only for the σ-additive case. Hence, for non additive set functions one might be interested to know what can be achieved through the constructive way. But even here a part of Fubini’s Theorem, namely that the integral with respect to the product of the set functions equals the repeated integral, remains valid only if the set function for the second integration is additive. More general, for submodular set functions one only gets an inequality: The repeated integral does not exceed the integral. Thus the problem of generalizing conditional expectation beyond the a-addtive case remains open to a large extent. For an alternative but likewise incomplete approach see Denneberg 1994. At least we prove in this chapter the full Theorem of Fubini for measures and also for (finitely) additive set functions.
Dieter Denneberg
Chapter 13. Representing Functionals as Integrals
Abstract
We have seen already how useful a representation theorem (Theorem 11.2) can be applied within the theory (for proving Corollary 11.3, Exercise 11.5 d) and Fubinis Theorem). As in Theorem 11.2 the crucial properties of a functional to be representable as an integral are mono-tonicity and comonotonic additivity (or, as in Greco 1982, a somewhat weaker condition). In Theorem 11.2 the domain of the functional is rather large. In decision situations one often has only restricted information, i.e. the domain of the functional is small. Representation theorems with minimal requirements on the domain are treated here. They are closely related to the extension theorems for set functions of Chapter 2. A further important question (e.g. in decision theory) 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.
Dieter Denneberg
Backmatter
Metadaten
Titel
Non-Additive Measure and Integral
verfasst von
Dieter Denneberg
Copyright-Jahr
1994
Verlag
Springer Netherlands
Electronic ISBN
978-94-017-2434-0
Print ISBN
978-90-481-4404-4
DOI
https://doi.org/10.1007/978-94-017-2434-0