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

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.

The convergence theorems we start with yield sufficient conditions that the integral can be interchanged with pointwise convergence of functions. These theorems clearly require continuity of the set function. First the Monotone Convergence Theorem derives from the special case proved in Chapter 1. It is valid very generally for monotone set functions. The other important theorem, Lebesgue’s Dominated Convergence Theorem, is first proved for the classical case of measures and later on it is generalized for subadditive set functions in applying the classical case with Lebesgue measure on the distribution functions. For this purpose we weaken pointwise convergence to stochastic convergence and further to convergence in distribution. These different types of convergence are important, too, in probability theory and statistics. By the way we get some information on measurability of the limit function even if the set function is not continuous.

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.

Convergence theorems for Choquet integrals with generalized autocontinuity

Participant noncompliance, in which participants do not follow their assigned treatment protocol, has long complicated the interpretation of randomized clinical trials. No gold standard has been identified for detecting noncompliance, but in some trials participants' biomarkers can provide objective information that suggests exposure to non-study treatments. However, existing methods are limited to retrospectively detecting noncompliance at a single time point based on a single biomarker measurement. We propose a novel method that can leverage participants' full biomarker history to detect noncompliance across multiple time points. Conditional on longitudinal biomarker data, our method can estimate the probability of compliance at (1) a single time point of the trial, (2) all time points, and (3) a future time point. Across time points, we model the biomarker as a mixture density with (latent) components corresponding to longitudinal patterns of compliance. To estimate the mixture density, we fit mixed effects models for both compliance and the biomarker. We use the mixture density to derive compliance probabilities that condition on the longitudinal biomarker data. We evaluate our compliance probabilities by simulation and apply them to a trial in which current smokers were asked to only smoke low nicotine study cigarettes (Center for the Evaluation of Nicotine in Cigarettes Project 1 Study 2). In the simulation, we investigated three different effects of compliance on the biomarker, as well as the effect of misspecification of the covariance structures. We compared probability estimators (1) and (2) to those that ignore the longitudinal correlation in the data according to area under the receiver operating characteristic curve. We evaluated estimator (3) by plotting its calibration lines. For Center for the Evaluation of Nicotine in Cigarettes Project 1 Study 2, we compared estimators (1) and (3) to a probability estimator of compliance at the last time point that ignores the longitudinal correlation. In the simulation, for both compliance at the last time point and at all time points, conditioning on the longitudinal biomarker data uniformly raised area under the receiver operating characteristic curve across all three compliance effect scenarios. The gains in area under the receiver operating characteristic curve were smaller under misspecification. The calibration lines for the prediction of compliance closely followed 45°, though with additional variability under misspecification. For compliance at the last time point of Center for the Evaluation of Nicotine in Cigarettes Project 1 Study 2, conditioning on participants' full biomarker history boosted area under the receiver operating characteristic curve by three percentage points. The prediction probabilities somewhat accurately approximated the non-longitudinal compliance probabilities. Compared to existing methods that only use a single biomarker measurement, our method can account for the longitudinal correlation in the biomarker and compliance to more accurately identify noncompliant participants. Our method can also use participants' biomarker history to predict compliance at a future time point.

Conditional expectations and submartingale sequences of random Schwartz distributions

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.

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 fuzzy integral on product spaces for NSA measures

Chapter 5 - Standard Convergence Theorems, The Fubini Theorem

Chapter 13 - The Henstock—Kurzweil Integral

The fuzzy integral