The fuzzy integral on product spaces for NSA measures

The fuzzy integral

Conditional expectations and submartingale sequences of random Schwartz distributions

In this paper we shall prove that the extension I of a generalized (or full) Daniell-integral I over a non distributive lattice E with values in a weakly б-distributive 1-group G has all the fundamental limiting properties of the Lebesgue integral, i.e. the properties stated i n the monotone convergence theorem, Fatou's-lemma and Lebesgue's dominated convergence theorem.

On the convergence of sequences of fuzzy measures and generalized convergence theorems of fuzzy integrals

This thesis is a study of a new area of analysis, quaternion measure theory, which is a generalization of positive measure theory, Since the quaternions H are non commutative, we need to define left intergration and right integration. In this thesis we can prove Lebesgue-Radon-Nikodym Theorem, Fubini Theorem, Lebesgue's Monotone Convergence Theorem, Lebesgue's Dominated Convergence Theorem and Riesz Representation Theorem for quaternion measures.

Measure theory is a branch of mathematics. Length, area, volume and weight are instances of measure concept. The emphasis in this chapter is mainly on the concept of measure, Borel set, measurable function, Lebesgue integral, Lebesgue-Stieltjes integral, monotone class theorem, Carathéodory extension theorem, measure continuity theorem, product measure theorem, monotone convergence theorem, Fatou’s lemma, Lebesgue dominated convergence theorem, and Fubini theorem. The main results in this chapter are well-known. For this reason the credit references are not given. This chapter can be omitted by the readers who are familiar with the basic concepts and theorems of measure and 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 Choquet integral in Riesz space

The convergence of (DG) fuzzy integrals

Some properties and convergence theorems of set-valued Choquet integrals

The main objective of this chapter is to remind the reader of some basic notion and fundamental facts about real analysis and functional analysis required for the comprehension of the following chapters. It is assumed that readers are familiar with the concept of metric and normed spaces. The definitions of a metric space, convergence of a sequence in a metric space, completeness, compactness and the Heine Borel theorem are introduced in the first part. Here there are some well–known properties of topological concepts needed to recall. Next the notion of norm and normed spaces, equivalent norms, compactness and relatively compactness, Banach space, dual space, weak and weak* convergence are presented. Before the Hilbert spaces, inner products and inner product spaces are briefly expressed. Here the relation between the Banach and Hilbert spaces is given by some examples. Most of the theorems of that part are stated without proof since they can easily be found from any real or functional analysis book given in the reference part. Following the Fatou lemma and the Lebesgue dominated convergence theorem, the chapter ends with some important theorems based on fixed point properties on Banach spaces. For instance the Tychonoff fixed point theorem is an extension of the Schauder’s fixed point theorem and the Schauder’s theorem is an extension of the Brouwer fixed point theorem. Since the proofs of these theorems require additional knowledge, we refer the reader to the book by Papageorgiou and Winkert [1] and reference therein. In addition [2–7] may also be functional.

Fuzzy number-valued triangular norm-based decomposable time-stamped fuzzy measure and integration

This section gives a summary of some elementary facts used frequently throughout this book, and can be regarded as an appendix. In particular, we consider sufficient conditions for the interchange of integration and limit operations. In detail, we discuss a result on uniform convergence, the dominated convergence theorem, the bounded convergence theorem, Fatou's lemma, and the monotone convergence theorem from the points of view of both Lebesgue integration theory and Riemann integration theory. Note that these are well-known results; hence we will be brief in details. For the proof of the monotone convergence theorem and Fubini's theorem we merely refer to the appropriate literature.

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.

Fuzzy measures and integrals defined on algebras of fuzzy subsets over complete residuated lattices