Convergence Theorems in the Theory of Integration

Extract This is the book we wish we had as graduate students. As its name suggests, this book is all about examples. Instead of listing a host of concepts all at once in an abstract setting, we bring ideas along slowly and illustrate each new idea with explicit and instructive examples. As one can see with the chapter titles, the focus of each chapter is on a specific operator and not on a concept. The important topics are covered through concrete operators and settings. As for style, we take great pains not to talk down to or above our audience. For example, we religiously eschew the dismissive words “obvious” and “trivial,” which have caused untold hours of heartache and self-doubt for puzzled graduate students the world over. Our prerequisites are minimal and we take time to highlight arguments and details that are often brushed over in other sources. In terms of prerequisites, we hope that the reader has had some exposure to Lebesgue’s theory of integration. Familiarity with the Lebesgue integral and the three big convergence theorems (Fatou’s lemma, the monotone convergence theorem, and the dominated convergence theorem) is sufficient for our purposes. In addition, an undergraduate-level course in complex analysis is needed for some of the chapters. We carefully develop everything else. Moreover, we cover any needed background material as part of the discussion. We do not burden the reader, who is anxious to get to operator theory, with a large volume of preliminary material. Nor do we make them pause their reading to chase down a concept or formula from an appendix.

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 Vitali convergence in measure theorem of nonlinear integrals

A measure and integration theory is presented in the quantum logic framework. A generalization of the monotone convergence theorem is proved. Counterexamples are used to show that the dominated convergence theorem, Fatou’s lemma, Egoroff’s theorem, and the additivity of the integral do not hold in this framework. Finally, a generalization of the Radon–Nikodym theorem is proved.

The Choquet integral in Riesz space

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.

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.

We define an integral of real-valued functions with respect to a measure that takes its values in the extended positive cone of a partially ordered vector space E. The monotone convergence theorem, Fatou’s lemma, and the dominated convergence theorem are established; the analogues of the classical \({\mathscr {L}}^1\)- and \({\mathrm L}^1\)-spaces are investigated. The results extend earlier work by Wright and specialise to those for the Lebesgue integral when E equals the real numbers. The hypothesis on E that is needed for the definition of the integral and for the monotone convergence theorem to hold (\(\sigma \)-monotone completeness) is a rather mild one. It is satisfied, for example, by the space of regular operators between a directed partially ordered vector space and a \(\sigma \)-monotone complete partially ordered vector space, and by every JBW-algebra. Fatou’s lemma and the dominated convergence theorem hold for every \(\sigma \)-Dedekind complete space. When E consists of the regular operators on a Banach lattice with an order continuous norm, or when it consists of the self-adjoint elements of a strongly closed complex linear subspace of the bounded operators on a complex Hilbert space, then the finite measures as in the current paper are precisely the strongly \(\sigma \)-additive positive operator-valued measures. When E is a partially ordered Banach space with a closed positive cone, then every positive vector measure is a measure in our sense, but not conversely. Even when a measure falls into both categories, the domain of the integral as defined in this paper can properly contain that of any reasonably defined integral with respect to the vector measure using Banach space methods.

In this chapter, we construct the Lebesgue integral of real-valued measurable functions with respect to a positive measure. After constructing the integral of measurable functions, we establish the three main convergence theorems, namely the monotone convergence theorem, Fatou’s lemma and the dominated convergence theorem. The last section gives typical applications to the continuity and differentiability of integrals of functions depending on a parameter. Important special cases of these applications are the Fourier transform and the convolution of functions.

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.

Conditional expectations and submartingale sequences of random Schwartz distributions

The fuzzy integral on product spaces for NSA measures

The fuzzy integral

Chapter 3 - The Lebesgue Integral

This chapter explains the concept of integration to measurable functions, which are approximated by simple functions to develop the Lebesgue integration theory. The chapter discusses the convergence properties of sequences of Lebesgue integrals and the approximation properties of Lebesgue integrable functions. Lebesgue extended the concept of integration to the space of measurable functions. The Lebesgue Dominated Convergence Theorem is the most frequently used theorem, which allows the operations of integration and limits to be interchanged. The Lebesgue integral is a generalization of the Riemann integral. Lusin's Theorem provides an approximation to a measurable function using continuous functions. The theory of measure and integration in the plane is very much analogous to that for the real line R . The fact that a double integral can be evaluated by iterated integration does not follow immediately from definition, but rather is a famous and difficult theorem called Fubini's Theorem. In practice, it is much easier to check the existence of iterated integrals instead of double integrals.