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Set Theoretical Aspects of Real Analysis
Set Theoretical Aspects of Real Analysis is built around a number of questions in real analysis and classical measure theory, which are of a set theoretic flavor. Accessible to graduate students, and researchers the beginning of the book presents introductory topics on real analysis and Lebesgue measure theory. These topics highlight the boundary between fundamental concepts of measurability and nonmeasurability for point sets and functions. The remainder of the book deals with more specialized material on set theoretical real analysis. The book focuses on certain logical and set theoretical aspects of real analysis. It is expected that the first eleven chapters can be used in a course on Le...
Measure Theory
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An Introduction to Sparse Stochastic Processes
A detailed guide to sparsity, providing a description of their transform-domain statistics and applying the models to practical algorithms.
Uniform Spaces and Measures
This book addresses the need for an accessible comprehensive exposition of the theory of uniform measures; the need that became more critical when recently uniform measures reemerged in new results in abstract harmonic analysis. Until now, results about uniform measures have been scattered through many papers written by a number of authors, some unpublished, written using a variety of definitions and notations. Uniform measures are certain functionals on the space of bounded uniformly continuous functions on a uniform space. They are a common generalization of several classes of measures and measure-like functionals studied in abstract and topological measure theory, probability theory, a...
Analysis in Banach Spaces
The present volume develops the theory of integration in Banach spaces, martingales and UMD spaces, and culminates in a treatment of the Hilbert transform, Littlewood-Paley theory and the vector-valued Mihlin multiplier theorem. Over the past fifteen years, motivated by regularity problems in evolution equations, there has been tremendous progress in the analysis of Banach space-valued functions and processes. The contents of this extensive and powerful toolbox have been mostly scattered around in research papers and lecture notes. Collecting this diverse body of material into a unified and accessible presentation fills a gap in the existing literature. The principal audience that we have in mind consists of researchers who need and use Analysis in Banach Spaces as a tool for studying problems in partial differential equations, harmonic analysis, and stochastic analysis. Self-contained and offering complete proofs, this work is accessible to graduate students and researchers with a background in functional analysis or related areas.
Convergence Theorems for Lattice Group-Valued Measures
Convergence Theorems for Lattice Group-valued Measures explains limit and boundedness theorems for measures taking values in abstract structures. The book begins with a historical survey about these topics since the beginning of the last century, moving on to basic notions and preliminaries on filters/ideals, lattice groups, measures and tools which are featured in the rest of this text. Readers will also find a survey on recent classical results about limit, boundedness and extension theorems for lattice group-valued measures followed by information about recent developments on these kinds of theorems and several results in the setting of filter/ideal convergence. In addition, each chapter ...
Positivity and Noncommutative Analysis
Capturing the state of the art of the interplay between positivity, noncommutative analysis, and related areas including partial differential equations, harmonic analysis, and operator theory, this volume was initiated on the occasion of the Delft conference in honour of Ben de Pagter's 65th birthday. It will be of interest to researchers in positivity, noncommutative analysis, and related fields. Contributions by Shavkat Ayupov, Amine Ben Amor, Karim Boulabiar, Qingying Bu, Gerard Buskes, Martijn Caspers, Jurie Conradie, Garth Dales, Marcel de Jeu, Peter Dodds, Theresa Dodds, Julio Flores, Jochen Glück, Jacobus Grobler, Wolter Groenevelt, Markus Haase, Klaas Pieter Hart, Francisco Hernánd...
Measure and Integration
This collection of Heinz König’s publications connects to his book of 1997 “Measure and Integration” and presents significant developments in the subject from then up to the present day. The result is a consistent new version of measure theory, including selected applications. The basic step is the introduction of the inner • (bullet) and outer • (bullet) premeasures and their extension to unique maximal measures. New “envelopes” for the initial set function (to replace the traditional Carathéodory outer measures) have been created, which lead to much simpler and more explicit treatment. In view of these new concepts, the main results are unmatched in scope and plainness, as ...
Reichenbach’s Paradise
Since its introduction by Hans Reichenbach, many philosophers have claimed to refute the idea – known as the common cause principle – that any surprising correlation between any two factors that do not directly influence one another is due to some common cause. For example, falsity of the principle is frequently inferred from falsifiability of Bell’s inequalities. The author demonstrates, however, that the situation is not so straightforward. There is more than one version of the principle formulated with the use of different variants of Reichenbach-inspired notions; their falsity still remains an open question. The book traces different formulations of the principle and provides proofs of a few pertinent theorems, settling the relevant questions in various probability spaces. In exploring mathematical and philosophical issues surrounding the principle, the book offers both philosophical insight and mathematical rigor.
