This book presents a unified treatise of the theory of measure and integration. In the setting of a general measure space, every concept is defined precisely and every theorem is presented with a clear and complete proof with all the relevant details. Counter-examples are provided to show that certain conditions in the hypothesis of a theorem cannot be simply dropped. The dependence of a theorem on earlier theorems is explicitly indicated in the proof, not only to facilitate reading but also to delineate the structure of the theory. The precision and clarity of presentation make the book an ideal textbook for a graduate course in real analysis while the wealth of topics treated also make the book a valuable reference work for mathematicians. The book is also very helpful to graduate students in statistics and electrical engineering, two disciplines that apply measure theory.
This volume consists of the proofs of 391 problems in Real Analysis: Theory of Measure and Integration (3rd Edition). Most of the problems in Real Analysis are not mere applications of theorems proved in the book but rather extensions of the proven theorems or related theorems. Proving these problems tests the depth of understanding of the theorems in the main text. This volume will be especially helpful to those who read Real Analysis in self-study and have no easy access to an instructor or an advisor.
Inhalt: Kurven - Reguläre Flächen - Die Geometrie der Gauß-Abbildung - Die innere Geometrie von Flächen - Anhang
The book is primarily devoted to the Kurzweil-Stieltjes integral and its applications in functional analysis, theory of distributions, generalized elementary functions, as well as various kinds of generalized differential equations, including dynamic equations on time scales. It continues the research that was paved out by some of the previous volumes in the Series in Real Analysis. Moreover, it presents results in a thoroughly updated form and, simultaneously, it is written in a widely understandable way, so that it can be used as a textbook for advanced university or PhD courses covering the theory of integration or differential equations.
With the success of its previous editions, Principles of Real Analysis, Third Edition, continues to introduce students to the fundamentals of the theory of measure and functional analysis. In this thorough update, the authors have included a new chapter on Hilbert spaces as well as integrating over 150 new exercises throughout. The new edition covers the basic theory of integration in a clear, well-organized manner, using an imaginative and highly practical synthesis of the "Daniell Method" and the measure theoretic approach. Students will be challenged by the more than 600 exercises contained in the book. Topics are illustrated by many varied examples, and they provide clear connections between real analysis and functional analysis. Gives a unique presentation of integration theory Over 150 new exercises integrated throughout the text Presents a new chapter on Hilbert Spaces Provides a rigorous introduction to measure theory Illustrated with new and varied examples in each chapter Introduces topological ideas in a friendly manner Offers a clear connection between real analysis and functional analysis Includes brief biographies of mathematicians
Besonderen Wert legt Rudin darauf, dem Leser die Zusammenhänge unterschiedlicher Bereiche der Analysis zu vermitteln und so die Grundlage für ein umfassenderes Verständnis zu schaffen. Das Werk zeichnet sich durch seine wissenschaftliche Prägnanz und Genauigkeit aus und hat damit die Entwicklung der modernen Analysis in nachhaltiger Art und Weise beeinflusst. Der "Baby-Rudin" gehört weltweit zu den beliebtesten Lehrbüchern der Analysis und ist in 13 Sprachen übersetzt. 1993 wurde es mit dem renommierten Steele Prize for Mathematical Exposition der American Mathematical Society ausgezeichnet. Übersetzt von Uwe Krieg.
The focus of this modern graduate text in real analysis is to prepare the potential researcher to a rigorous "way of thinking" in applied mathematics and partial differential equations. The book will provide excellent foundations and serve as a solid building block for research in analysis, PDEs, the calculus of variations, probability, and approximation theory. All the core topics of the subject are covered, from a basic introduction to functional analysis, to measure theory, integration and weak differentiation of functions, and in a presentation that is hands-on, with little or no unnecessary abstractions. Additional features: * Carefully chosen topics, some not touched upon elsewhere: fine properties of integrable functions as they arise in applied mathematics and PDEs – Radon measures, the Lebesgue Theorem for general Radon measures, the Besicovitch covering Theorem, the Rademacher Theorem; topics in Marcinkiewicz integrals, functions of bounded variation, Legendre transform and the characterization of compact subset of some metric function spaces and in particular of Lp spaces * Constructive presentation of the Stone-Weierstrass Theorem * More specialized chapters (8-10) cover topics often absent from classical introductiory texts in analysis: maximal functions and weak Lp spaces, the Calderón-Zygmund decomposition, functions of bounded mean oscillation, the Stein-Fefferman Theorem, the Marcinkiewicz Interpolation Theorem, potential theory, rearrangements, estimations of Riesz potentials including limiting cases * Provides a self-sufficient introduction to Sobolev Spaces, Morrey Spaces and Poincaré inequalities as the backbone of PDEs and as an essential environment to develop modern and current analysis * Comprehensive index This clear, user-friendly exposition of real analysis covers a great deal of territory in a concise fashion, with sufficient motivation and examples throughout. A number of excellent problems, as well as some remarkable features of the exercises, occur at the end of every chapter, which point to additional theorems and results. Stimulating open problems are proposed to engage students in the classroom or in a self-study setting.
This is part two of a two-volume book on real analysis and is intended for senior undergraduate students of mathematics who have already been exposed to calculus. The emphasis is on rigour and foundations of analysis. Beginning with the construction of the number systems and set theory, the book discusses the basics of analysis (limits, series, continuity, differentiation, Riemann integration), through to power series, several variable calculus and Fourier analysis, and then finally the Lebesgue integral. These are almost entirely set in the concrete setting of the real line and Euclidean spaces, although there is some material on abstract metric and topological spaces. The book also has appendices on mathematical logic and the decimal system. The entire text (omitting some less central topics) can be taught in two quarters of 25–30 lectures each. The course material is deeply intertwined with the exercises, as it is intended that the student actively learn the material (and practice thinking and writing rigorously) by proving several of the key results in the theory.
An Introduction to Measure-Theoretic Probability, Second Edition, employs a classical approach to teaching the basics of measure theoretic probability. This book provides in a concise, yet detailed way, the bulk of the probabilistic tools that a student working toward an advanced degree in statistics, probability and other related areas should be equipped with. This edition requires no prior knowledge of measure theory, covers all its topics in great detail, and includes one chapter on the basics of ergodic theory and one chapter on two cases of statistical estimation. Topics range from the basic properties of a measure to modes of convergence of a sequence of random variables and their relationships; the integral of a random variable and its basic properties; standard convergence theorems; standard moment and probability inequalities; the Hahn-Jordan Decomposition Theorem; the Lebesgue Decomposition T; conditional expectation and conditional probability; theory of characteristic functions; sequences of independent random variables; and ergodic theory. There is a considerable bend toward the way probability is actually used in statistical research, finance, and other academic and nonacademic applied pursuits. Extensive exercises and practical examples are included, and all proofs are presented in full detail. Complete and detailed solutions to all exercises are available to the instructors on the book companion site. This text will be a valuable resource for graduate students primarily in statistics, mathematics, electrical and computer engineering or other information sciences, as well as for those in mathematical economics/finance in the departments of economics. Provides in a concise, yet detailed way, the bulk of probabilistic tools essential to a student working toward an advanced degree in statistics, probability, and other related fields Includes extensive exercises and practical examples to make complex ideas of advanced probability accessible to graduate students in statistics, probability, and related fields All proofs presented in full detail and complete and detailed solutions to all exercises are available to the instructors on book companion site Considerable bend toward the way probability is used in statistics in non-mathematical settings in academic, research and corporate/finance pursuits.
Probability and Mathematical Statistics: A Series of Monographs and Textbooks: A Graduate Course in Probability presents some of the basic theorems of analytic probability theory in a cohesive manner. This book discusses the probability spaces and distributions, stochastic independence, basic limiting operations, and strong limit theorems for independent random variables. The central limit theorem, conditional expectation and martingale theory, and Brownian motion are also elaborated. The prerequisite for this text is knowledge of real analysis or measure theory, particularly the Lebesgue dominated convergence theorem, Fubini's theorem, Radon-Nikodym theorem, Egorov's theorem, monotone convergence theorem, and theorem on unique extension of a sigma-finite measure from an algebra to the sigma-algebra generated by it. This publication is suitable for a one-year graduate course in probability given in a mathematics program and preferably for students in their second year of graduate work.
Dieser Buchtitel ist Teil des Digitalisierungsprojekts Springer Book Archives mit Publikationen, die seit den Anfängen des Verlags von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv Quellen für die historische wie auch die disziplingeschichtliche Forschung zur Verfügung, die jeweils im historischen Kontext betrachtet werden müssen. Dieser Titel erschien in der Zeit vor 1945 und wird daher in seiner zeittypischen politisch-ideologischen Ausrichtung vom Verlag nicht beworben.
Dieses Lehr- und Handbuch behandelt sowohl die elementaren Konzepte als auch die fortgeschrittenen und zukunftsweisenden linearen und nichtlinearen FE-Methoden in Statik, Dynamik, Festkörper- und Fluidmechanik. Es wird sowohl der physikalische als auch der mathematische Hintergrund der Prozeduren ausführlich und verständlich beschrieben. Das Werk enthält eine Vielzahl von ausgearbeiteten Beispielen, Rechnerübungen und Programmlisten. Als Übersetzung eines erfolgreichen amerikanischen Lehrbuchs hat es sich in zwei Auflagen auch bei den deutschsprachigen Ingenieuren etabliert. Die umfangreichen Änderungen gegenüber der Vorauflage innerhalb aller Kapitel - vor allem aber der fortgeschrittenen - spiegeln die rasche Entwicklung innerhalb des letzten Jahrzehnts auf diesem Gebiet wieder. TOC:Eine Einführung in den Gebrauch von Finite-Elemente-Verfahren.-Vektoren, Matrizen und Tensoren.-Einige Grundbegriffe ingenieurwissenschaftlicher Berechnungen.-Formulierung der Methode der finiten Elemente.-Formulierung und Berechnung von isoparametrischen Finite-Elemente-Matrizen.-Nichtlineare Finite-Elemente-Berechnungen in der Festkörper- und Strukturmechanik.-Finite-Elemente-Berechnungen von Wärmeübertragungs- und Feldproblemen.-Lösung von Gleichgewichtsbeziehungen in statischen Berechnungen.-Lösung von Bewegungsgleichungen in kinetischen Berechnungen.-Vorbemerkungen zur Lösung von Eigenproblemen.-Lösungsverfahren für Eigenprobleme.-Implementierung der Finite-Elemente-Methode.
Measure, Integration, and Functional Analysis deals with the mathematical concepts of measure, integration, and functional analysis. The fundamentals of measure and integration theory are discussed, along with the interplay between measure theory and topology. Comprised of four chapters, this book begins with an overview of the basic concepts of the theory of measure and integration as a prelude to the study of probability, harmonic analysis, linear space theory, and other areas of mathematics. The reader is then introduced to a variety of applications of the basic integration theory developed in the previous chapter, with particular reference to the Radon-Nikodym theorem. The third chapter is devoted to functional analysis, with emphasis on various structures that can be defined on vector spaces. The final chapter considers the connection between measure theory and topology and looks at a result that is a companion to the monotone class theorem, together with the Daniell integral and measures on topological spaces. The book concludes with an assessment of measures on uncountably infinite product spaces and the weak convergence of measures. This book is intended for mathematics majors, most likely seniors or beginning graduate students, and students of engineering and physics who use measure theory or functional analysis in their work.
An introduction to analysis with the right mix of abstract theories and concrete problems. Starting with general measure theory, the book goes on to treat Borel and Radon measures and introduces the reader to Fourier analysis in Euclidean spaces with a treatment of Sobolev spaces, distributions, and the corresponding Fourier analysis. It continues with a Hilbertian treatment of the basic laws of probability including Doob's martingale convergence theorem and finishes with Malliavin's "stochastic calculus of variations" developed in the context of Gaussian measure spaces. This invaluable contribution gives a taste of the fact that analysis is not a collection of independent theories, but can be treated as a whole.
Der vorliegende Band stellt den zweiten Teil eines Analysis-Kurses für Studierende der Mathematik und Physik dar. Das erste Kapitel über Differentialrechnung im R^n behandelt nach einer Einführung in die topologischen Grundbegriffe Kurven im R^n, partielle Ableitungen, totale Differenzierbarkeit, Taylorsche Formel, Maxima und Minima von Funktionen mehrerer Veränderlichen, implizite Funktionen und parameterabhängige Integrale. Das zweite Kapitel gibt eine kurze Einführung in die Theorie der gewöhnlichen Differentialgleichungen. Nach dem Beweis des allgemeinen Existenz- und Eindeutigkeitssatzes und der Besprechung der Methode der Trennung der Variablen wird besonders auf die Theorie der linearen Differentialgleichungen eingegangen.
In den Bachelor-Studiengängen der Mathematik steht für die Komplexe Analysis (Funktionentheorie) oft nur eine einsemestrige 2-stündige Vorlesung zur Verfügung. Dieses Buch eignet sich als Grundlage für eine solche Vorlesung im 2. Studienjahr. Mit einer guten thematischen Auswahl, vielen Beispielen und ausführlichen Erläuterungen gibt dieses Buch eine Darstellung der Komplexen Analysis, die genau die Grundlagen und den wesentlichen Kernbestand dieses Gebietes enthält. Das Buch bietet über diese Grundausbildung hinaus weiteres Lehrmaterial als Ergänzung, sodass es auch für eine 3- oder 4 –stündige Vorlesung geeignet ist. Je nach Hörerkreis kann der Stoff unterschiedlich erweitert werden. So wurden für den „Bachelor Lehramt“ die geometrischen Aspekte der Komplexen Analysis besonders herausgearbeitet.
Dieses Buch ist der erste Teil eines zweibändigen Werkes über Analysis. Es ist aus Vorlesungen, Übungen und Seminaren erwachsen, die ich mehrfach an den Universitäten Mainz und Karlsruhe gehalten habe, und so angelegt, daß es auch zum Selbststudium dienen kann. Ich widerstehe der Versuchung, dem Studenten, der jetzt dieses Vorwort liest, ausführlich die Themen zu beschreiben, die ihn erwarten; denn dazu müßte ich Worte gebrauchen, die er doch erst nach der Lektüre des Buches verstehen kann - nach der Lektüre aber sollte er selbst wissen, was gespielt worden ist. Den Kenner hingegen wird ein Blick auf das Inhaltsverzeichnis und ein rasches Durchblättern ausreichend orientieren. Dennoch halte ich es für möglich, anknüpfend an Schulkenntnisse und Alltagser fahrung auch dem Anfänger verständlich zu machen, was der rote Faden ist, der dieses Buch durchzieht und in welchem Geist es geschrieben wurde und gelesen werden möchte. Der rote Faden, das ständig aufklingende Leitmotiv und energisch vorwärts treibende Hauptproblem ist die Frage, wie man das Änderungsverhalten einer Funktion verstehen, beschreiben und beherrschen kann, schärfer: Welche Be griffe eignen sich am besten dazu, die Änderung einer Funktion "im Kleinen" (also bei geringen Änderungen ihrer unabhängigen Variablen) zu erfassen, was kann man über die Funktion "im Großen", über ihren Gesamtverlauf sagen, wenn man Kenntnisse über ihr Verhalten "im Kleinen" hat, geben uns diese Kenntnisse vielleicht sogar die Funktion gänzlich in die Hand odq besser: Wie tief müssen diese "lokalen Kenntnisse" gehen, um uns die Funktion "global"
This book covers the subject matter that is central to mathematical analysis: measure & integration theory, some point set topology, & rudiments of functional analysis. Also, a number of other topics are developed to illustrate the uses of this core material in important areas of mathematics & to introduce readers to more advanced techniques. Some of the material presented has never appeared outside of advanced monographs & research papers, or been readily available in comparative texts. About 460 exercises, at varying levels of difficulty, give readers practice in working with the ideas presented here.

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