Integration of Measurable 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.

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.

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.

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.

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.

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-valued integrals of fuzzy-valued measurable functions with respect to fuzzy-valued measures based on closed intervals

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.

Conditional expectations and submartingale sequences of random Schwartz distributions

The fuzzy integral on product spaces for NSA measures

The fuzzy integral

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.

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.

In the theories of Lebesgue integration and of ordinary differential equations, the Lebesgue Dominated Convergence Theorem provides one of the most widely used tools. Available analogy in the Riemann or Riemann–Stieltjes integration is the Bounded Convergence Theorem, sometimes called also the Arzelà or Arzelà–Osgood or Osgood Theorem. In the setting of the Kurzweil–Stieltjes integral for real valued functions its proof can be obtained by a slight modification of the proof given for the $$\sigma $$ -Young–Stieltjes integral by T.H. Hildebrandt in his monograph from 1963. However, it is clear that the Hildebrandt’s proof cannot be extended to the case of Banach space-valued functions. Moreover, it essentially utilizes the Arzelà Lemma which does not fit too much into elementary text-books. In this paper, we present the proof of the Bounded Convergence Theorem for the abstract Kurzweil–Stieltjes integral in a setting elementary as much as possible.