Convergence theorems for Choquet integrals with generalized autocontinuity

There are several types of nonlinear integrals with respect to nonadditive measures, such as the Choquet, Sipos, Sugeno, and Shilkret integrals. In order to put those integrals into practical use and aim for application to various fields, it is indispensable to establish convergence theorems of such nonlinear integrals. However, they have individually been discussed for each of the integrals up to the present. In this article, several important convergence theorems of nonlinear integrals, such as the monotone convergence theorem, the bounded convergence theorem, and the Vitali convergence theorem, are formulated in a unified way regardless of the types of integrals.

Choquet integrals of set-valued functions with respect to set-valued fuzzy measures

Some properties and convergence theorems of set-valued Choquet integrals

The Vitali convergence in measure theorem of nonlinear integrals

The fuzzy integral

An elementary application of Fatou’s lemma gives a strengthened version of the monotone convergence theorem. We call this the convergence from below theorem. We make the case that this result should be better known, and deserves a place in any introductory course on measure and integration. 1 The convergence from below theorem Three famous convergence-related results appear in most introductory courses on measure and integration: the monotone convergence theorem, Fatou’s lemma and the dominated convergence theorem. In teaching this material it is common to follow the approach taken in, for example, [1, Chapter 1]. There Rudin begins by proving the monotone convergence theorem and then deduces Fatou’s lemma. Finally, he deduces the dominated convergence theorem from Fatou’s lemma. The result which we call the convergence from below theorem (Theorem 1.2 below) is essentially distilled from this proof of the dominated convergence theorem ([1, pp. 26-27]). We do not claim originality for this result, or for the related Theorem 1.3. They are presumably known, although we know of no explicit references for them. However, we wish to make a case that that they should be better known than they are. In particular, we suggest that Theorem 1.2 deserves a name and a place in the syllabus when this material is taught. Throughout we discuss results concerning pointwise convergence. In the usual way, there are versions of all these results in terms of almost-everywhere convergence instead. For convenience, we shall use the following terminology. Let X be a set, let (fn) be a sequence of functions from X to [0,∞] and let f be another function from X to [0,∞]. We say that the functions fn converge to f from below on X if the functions fn tend to f pointwise on X and fn(x) ≤ f(x) (n ∈ N, x ∈ X). We say that the functions fn converge to f monotonely from below on X if the functions fn tend to f pointwise on X and, for all x ∈ X, we have f1(x) ≤ f2(x) ≤ f3(x) ≤ · · ·. We begin by recalling the statement of the monotone convergence theorem.

The interchange of the ‘limit of an integral’ with the ‘integral of a limit’ for sequenc- es of functions is crucial in relevant applications, such as Fourier series for decom- posing periodic functions into sinusoidal components, and Fubini’s theorem for changing the order of integration of multivariable functions. This expository paper reviews three classical results in real analysis for cases where the limit of an integral of a sequence of functions equals the integral of the limiting function: (1) Mono- tone Convergence Theorem, (2) Uniform Convergence Theorem, and the broad- est result, (3) Dominated Convergence Theorem. While proofs of (2) are typically studied in undergraduate analysis, the proofs of (1) and (3) are usually reserved for graduate-level measure theory, where they are taught in a more general context. The purpose of this paper is to summarize and adapt W. A. J. Luxembourg’s un- dergraduate-friendly proof [7] of (3) Arzel`a’s Dominated Convergence Theorem, to demonstrate the nontrivial direction of (1) Monotone Convergence Theorem for Riemann Integrals. Our aim is to demystify the hidden logic involved in these well-established theorems, making them more accessible for undergraduate analysis.

It is well known that the Choquet and Sugeno integrals w.r.t. a discrete fuzzy measure can be considered as aggregation operators. In this paper, we study in detail two special cases of symmetric fuzzy measures, i.e. fuzzy measures of which the values only depend upon the cardinality of the arguments. The first case is that of a symmetric k-order additive fuzzy measure, i.e. a fuzzy measure of which the Möbius transform vanishes in sets with cardinality greater than k. A new increasing sequence of binomial OWA operators is introduced. It is recalled that weighted sums of aggregation operators, with possibly negative weights, may also lead to aggregation operators. The Choquet integral w.r.t. a symmetric k-order additive fuzzy measure is then characterized as such a weighted sum of the first k binomial OWA operators. The second case is that of a symmetric k-order maxitive fuzzy measure, i.e. a fuzzy measure of which the possibilistic Möbius transform vanishes in sets with cardinality greater than k. The Sugeno integral w.r.t. a symmetric k-order maxitive fuzzy measure is then characterized as a weighted maximum of the last k order statistics.

The fuzzy integral on product spaces for NSA measures

Convergence theorems for random elements in convex combination spaces

Markovian imprecise jump processes: Extension to measurable variables, convergence theorems and algorithms

Fuzzy measure shift differentiation of the Choquet integral for a nonnegative measurable function taken with respect to a fuzzy measure over a real fuzzy measure space is proposed. It is applied to financial engineering. First, a real interval limited Choquet integral for a nonnegative measurable function taken with respect to a fuzzy measure over a real fuzzy measure space is given, then a fuzzy measure left shift differential coefficient, a fuzzy measure right shift differential coefficient, a fuzzy measure shift differential coefficient, and a fuzzy measure shift derived function of the real interval limited Choquet integral for a nonnegative measurable function over a real fuzzy measure space along the domain are defined by the limitation process of a fuzzy measure shift. Two examples of a fuzzy measure shift differentiation are given, where fuzzy measure distributions are either a continuous distribution or a discrete distribution, to understand the notion of the fuzzy measure shift differentiation. Moreover, they are applied to financial option trading. The pricing models of a European call option premium and a European put option premium are defined using the real interval limited Choquet integral for a nonnegative measurable function over a real fuzzy measure space. Then, the distribution of underlying securities of an option trading at the expiration date is given as a /spl lambda/-fuzzy measure, where the total fuzzy measure is equal to one. An important risk index, the delta, which is the rate of change of the premium with respect to underlying security price is defined using the fuzzy measure shift differentiation of the real interval limited Choquet integral for a nonnegative measurable function over a real fuzzy measure space. Finally, these option trading models based on the real interval limited Choquet integral over a real fuzzy measure space is tested with the real market data and is compared with the popular option trading model based on the probability measure and logarithmic normal distribution defined by Black and Sholes (1973).

The Choquet integral with respect to nonadditive monotone set functions, including imprecise probabilities and fuzzy measures, is a generalization of the classical Lebesgue integral. It is one kind of nonlinear functionals defined on a subspace of all real valued measurable functions. In this paper, several different types of convergence, including the mean convergence that is based on the Choquet integral, for sequences of measurable functions are considered, and the corresponding convergence theorems for sequence of Choquel integrals are demonstrated. Particularly, the theorem of convergence in measure is presented in a form of “necessary and sufficient condition” by using the structural characteristics of nonnegative monotone set functions. As an application of convergence theorems, the stability of a class of nonlinear integral systems is discussed.

Chapter 5 - Standard Convergence Theorems, The Fubini Theorem

Chapter 13 - The Henstock—Kurzweil Integral