Measure and Integral

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.

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.

Chapter 3 - The Lebesgue Integral

The fuzzy integral on product spaces for NSA measures

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.

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.

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.

The Dominated Convergence Theorem, Fatou's Lemma, and the Monotone Convergence Theorem are collectively referred to as the continuity theorems of Lebesgue integration. These ubiquitous results are stated and proved in the literature for sequences of measurable functions. It is well known that these results do not hold for arbitrary nets of measurable functions. These results, as well as the Levy Continuity Theorem, Riesz's characterization of $\mathcal{L}_p$ convergence in terms of almost everywhere convergence and convergence of norms, and a stronger version of Scheffe's Theorem, are derived for nets indexed by countably accessible directed sets.

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.

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 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.

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.

In this chapter we define and study the Lebesgue integral of functions on \(\mathbb {R}^d\) (or on subsets of \(\mathbb {R}^d\)). We first define the Lebesgue integral for nonnegative functions in Section 4.1, and in Section 4.2 prove two fundamental results on convergence of integrals: Fatou’s Lemma and the Monotone Convergence Theorem. We define the integral of extended real-valued and complex-valued functions in Section 4.3. Integrable functions (those functions for which the integral of |f| is finite) are introduced in Section 4.4, as is the Lebesgue space \(L^1(E),\) which is the set of all integrable functions on E. In Section 4.5 we prove the Dominated Convergence Theorem, or DCT, which is one of the most useful theorems in analysis. In particular, we use the DCT to show that integrable functions can be well-approximated by a wide variety of functions that have special properties, including simple functions, continuous functions, and step functions. Among other applications, this allows us to characterize Riemann integrable functions as the functions that are continuous at almost every point, and to establish the relationship between Lebesgue and Riemann integrals. Finally, Section 4.6 covers the important theorems of Fubini and Tonelli, which tell us when we can exchange the order of iterated integrals.

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.

This chapter develops the basic theory of measure and integration, including Lebesgue measure and Lebesgue integration for the line. Section 1 introduces measures, including 1-dimensional Lebesgue measure as the primary example, and develops simple properties of them. Sections 2–4 introduce measurable functions and the Lebesgue integral and go on to establish some easy properties of integration and the fundamental theorems about how Lebesgue integration behaves under limit operations. Sections 5–6 concern the Extension Theorem announced in Section 1 and used as the final step in the construction of Lebesgue measure. The theorem allows $\sigma$-finite measures to be extended from algebras of sets to $\sigma$-algebras. The theorem is proved in Section 5, and the completion of a measure space is defined in Section 6 and related to the proof of the Extension Theorem. Section 7 treats Fubini's Theorem, which allows interchange of order of integration under rather general circumstances. This is a deep result. As part of the proof, product measure is constructed and important measurability conditions are established. This section mentions that Fubini's Theorem will be applicable to higher-dimensional Lebesgue measure, but the details are deferred to Chapter VI. Section 8 extends Lebesgue integration to complex-valued functions and to functions with values in finite-dimensional vector spaces. Section 9 gives a careful definition of the spaces $L^1$, $L^2$, and $L^{\infty}$ for any measure space, introduces the notion of a normed linear space, and verifies that these three spaces are examples. The main theorem of the section about $L^1$, $L^2$, and $L^{\infty}$ is the completeness of these three spaces as metric spaces. In addition, the section proves a version of Alaoglu's Theorem concerning weak-star convergence.