Product Description
This textbook and treatise begins with classical real variables, develops the Lebesgue theory abstractly and for Euclidean space, and analyzes the structure of measures. The authors' vision of modern real analysis is seen in their fascinating historical commentary and perspectives with other fields. There are comprehensive treatments of the role of absolute continuity, the evolution of the Riesz representation theorem to Radon measures and distribution theory, weak convergence of measures and the Dieudonné-Grothendieck theorem, modern differentiation theory, fractals and self-similarity, rearrangements and maximal functions, and surface and Hausdorff measures. There are hundreds of illuminating exercises, and extensive, focused appendices on functional and Fourier analysis. The presentation is ideal for the classroom, self-study, or professional reference. ... Read more

Product Description
Presenting the various approaches to the study of integration, well-known mathematics professor Angus E. Taylor brings together in one volume "a blend of the particular and the general, of the concrete and the abstract"—an invaluable study aid for beginning graduate students. 38 diagrams. Introduction. List of Special Symbols. Index.
... Read more

Customer Reviews (1)

Gotta Be Nuts To Use Anything Else.
I have many friends who are graduate students or upper level undergrads in mathematics at Columbia and Yale who are suffering in their Real Variables courses because their arrogant,aloof instructors are using overpriced,ultra-TERSE higher analysis texts like Rudin's Real And Complex Analysis or Royden's incomprehensibly "classic" text.(For those of you who are using Royden and spent the insane cover price on it,you should realize there are a TON of errors in it.)I feel so bad for them because they really got taken for a ride-for a fraction of the cost,they could have had this GEM of a text.Taylor,famous for his Advanced Calculus,developed this text as a series of lecture notes at UCLA in the 1950's.Taylor rightly realizes the key to understanding mathematical analysis/measure theory at this level is a firm grip of point-set theory and abstract topologicalspaces-of which the real and complex numbers systems are but special cases.He then proceeds to develop the metric and topological properties of the number systems in this abstract setting-focusing on the rigorous set-theoretic foundations. Critical to analysis(and pure math in general) are PRECISE DEFINITIONS-and this is what makes Taylor's approach to the subject so beautiful and crystal clear,without sacrificingrigor or exposition of challenging theories.How many analysis text do you know develop BOTH the classical Lesbegue approachto modern integration as well as the Daniell approach for functional analysis-without losing an ounce of clarity?Run-don't-walk to your bookstore and order this if you're in upper level analysis courses. This will become a standard reference throughout you're career-I promise. ... Read more

Product Description

Also called Ito calculus, the theory of stochastic integration has applications in virtually every scientific area involving random functions. This introductory textbook provides a concise introduction to the Ito calculus.

From the reviews:

"Introduction to Stochastic Integration is exactly what the title says. I would maybe just add a ‘friendly’ introduction because of the clear presentation and flow of the contents." --THE MATHEMATICAL SCIENCES DIGITAL LIBRARY

Customer Reviews (3)

Very good!!!
Highly recommend this book to everyone who started to study stochastic processes and SDE! This book gives better understanding and intuition of the subject than more advanced Karatzas & Shreve. I enjoyed to read this book very much also because the author always referees you to the necessary formula/theorem/definition that was previously given in the text. So, there is no need to search all over the book trying to guess why the author made a current step.

The best introduction to stochatic integration ever!!!
Personally, I think this is the best introduction to stochastic integration ever! The author did a remarkable job in presenting the Ito calculus and SDE to readers in an extremely clear way. The author always motivates the readers with intuitive thinking, then leads them to rigorous theory followed by contrete examples. The book only assumes some knowledge of measure theory and is accessible to a wide audience. However, no mathematical rigor is sacrificed. The author has amazingly achieved clarity, rigor and readability in a single book.

The author keeps audience in mind all the time. He never assumes you may know some result used for proof. He is even so generous in spending a subsection on Borel-Cantelli Lemma and ChebySheve Inequality. So the book is really self-contained! You will find the transition between any two sections or subsections very smooth too.

Besides the standard topics you may find in an introductory book in this fields such as Ito integral, Ito Formula, Stratonovich Integral, Tanaka's Formula, Local Time and Girsanov Theorem, SDE and applications, the author also gives careful treatment of Wiener integral to bridge the gap between Stieltjes integral and Ito Integral. (Compensated) Poisson process is also thoroughly covered in many examples, although you may not see this in the content list.

A lot of exercises are given at the end of each chapter. I consider this a great gift for readers. Many textbooks on this topic put the stuff that cannot be covered in maintext into exercises or simply stack problems from other books as exercises. This book is different. The exercises are carefully designed to help readers reinforce what they've learned. So most of the exercises can be tackled by a serious graduate student in math.

In a word, this is the best introductory textbook on this hard subject I've ever seen. I believe even an expert will find useful and interesting material in it. It definitely deserves a 5-star rating!

An excellent text
From perspective of real analysis, this text gives a rigorous introduction to Ito calculus. In an extremely smooth fashion, all chapters flow out one by one. A background on real analysis is required. However, is it true that some basics of real analysis, e.g. Riemann integration, measurability, etc, are prerequisites for stochastic calculus? It was a wonderful experience for me to read though the book. ... Read more

Product Description
This book presents a historical development of the integration theories of Riemann, Lebesgue, Henstock-Kurzweil, and McShane, showing how new theories of integration were developed to solve problems that earlier theories could not handle. It develops the basic properties of each integral in detail and provides comparisons of the different integrals. The chapters covering each integral are essentially independent and can be used separately in teaching a portion of an introductory course on real analysis. There is a sufficient supply of exercises to make the book useful as a textbook. ... Read more

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The first edition of Hardy's Integration of Functions of a Single Variable was published in 1905, with this 1916 second edition being reprinted up until 1966. Now this digital reprint of the second edition will allow the twenty-first-century reader a fresh exploration of the text. Hardy's chapters provide a comprehensive review of elementary functions and their integration, the integration of algebraic functions and Laplace's principle, and the integration of transcendental functions. The text is also saturated with explanatory notes and usable examples centred around the elementary problem of indefinite integration and its solutions. Appendices contain useful bibliographic references and a workable demonstration of Abel's proof, rewritten specifically for the second edition. This innovative tract will continue to be of interest to all mathematicians specialising in the theory of integration and its historical development. ... Read more

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It has been 13 years since the first edition of Stochastic Integration and Differential Equations, A New Approach appeared, and in those years many other texts on the same subject have been published, often with connections to applications, especially mathematical finance. Yet in spite of the apparent simplicity of approach, none of these books has used the functional analytic method of presenting semimartingales and stochastic integration. Thus a 2nd edition seems worthwhile and timely, though we will no longer call it"a new approach."

The new edition has several significant changes, most prominently the addition of exercises for solution. These are intended to supplement the text, but lemmas needed in a proof are never relegated to the exercises! Many of the exercises have been tested by graduate students at Purdue and Cornell Universities. Chap. 3 has been nearly completely redone, with a new, more intuitive and simultaneously elementary proof of the fundamental Doob-Meyer decomposition theorem, the more general version of the Girsanov theorem due to Lenglart, the Kazamaki-Novikov criteria for exponential local martingales to be martingales, and a modern treatment of compensators. Chap. 4 treats sigma martingales (important in finance theory) and gives a more comprehensive treatment of martingale representation, including both the Jacod-Yor theory and Emery's examples of martingales that actually have martingale representation (thus going beyond the standard cases of Brownian motion and the compensated Poisson process). New topics added include an introduction to the theory of the expansion of filtrations, and an elementary treatment of the Burkholder-Gundy-Fefferman martingale inequalities. Last, there are of course small changes throughout the book. ... Read more

Customer Reviews (2)

Received a cheaper knock-off printing
This review is about the material quality of the printing in the copy I received. This is not about the content.

I have access to a copy of this second edition (third corrected printing, 2005) in my office. It is the usual Springer high quality hardcover: it has a matte cover with texture, beautifully bound; the paper inside is high-quality, very soft and slightly off-white; and the printing of the text is very sharp. The version I received from Amazon claimed to be exactly the same, but was very different:
- The hardcover was shiny, did not have texture, and had a natural tendency to bend strongly outwards;
- The paper inside is whiter, horribly white;
- The text printing looks like a photocopy of the original.
It looks and feels like a cheap knock-off photocopy done in a backyard. They didn't even bother changing the information about the printing including the 'Printed in Germany' and 'Printed on acid-free paper'.

When I pay a lot of money for a hardcover edition I want the real thing, not a cheap knock-off. I returned mine.

Very good textbook
This book covers EXACTLY what the title promises. The approach to stochastic integration is a bit different from other books I have read, but very intuitive, general and understandable.
The proofs of the theorems are generally quite easy to understand, but do not lack rigor at all - quite the opposite.
The prerequisites in probability theory are quite modest, however some understanding of measure theory (at least on an intuitive level) is helpful. Some knowledge of stochastic processes (such as maertingales...) also facilitates the reading of the book.

Overall, it is quite amazing how Prof. Protter manages to explain deep subjects rigorously, yet very understandably!

The book contains a "normal" amount of typos, but almost all are harmless - they do not affect the understanding at all.
The only draw-back to the serious researcher might be the large number of French language references. ... Read more

Product Description

This textbook introduces geometric measure theory through the notion of currents. Currents, continuous linear functionals on spaces of differential forms, are a natural language in which to formulate types of extremal problems arising in geometry, and can be used to study generalized versions of the Plateau problem and related questions in geometric analysis.

The text provides considerable background for the student and discusses techniques that are applicable to complex geometry, partial differential equations, harmonic analysis, differential geometry, and many other parts of mathematics. Topics include the deformation theorem, the area and coareas formulas, the compactness theorem, the slicing theorem and applications to minimal surfaces.

Motivating key ideas with examples and figures, Geometric Integration Theory is a comprehensive introduction ideal for both use in the classroom and for self-study. The exposition demands minimal background, is self-contained and accessible, and thus is ideal for both graduate students and researchers.

Product Description
This introductory text acts as a singular resource for undergraduates learning the fundamental principles and applications of integration theory.Chapters discuss: function spaces and functionals, extension of Daniell spaces, measures of Hausdorff spaces, spaces of measures, elements of the theory of real functions on R. ... Read more

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The fundamental concept of Lebesgue integration provides a basis for approaching theoretical and applied problems in mathematics, engineering, physics and many other sciences. Lebesgue Integration presents a logical development of the basic concepts of Lebesgue's integration theorems, proceeding from the study of topological concepts on the real line. This text is intended for advanced undergraduates and beginning graduate students and is also suitable for self-study by students preparing for the masters or doctoral preliminary examination. ... Read more

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Customer Reviews (3)

Fatal inaccuracies
There's a stupid mistake in the first set of examples on page 16: 1 (vi) should be subset not equality. He also uses seriously non-standard notation for sets and spaces and so on. Don't let it slow you down.

Something new in measure theory here
The book isn't organized as a teaching text: is is a definition, theorem and proof type structure that is pretty hard reading. I would have given it two stars except for the convergence diagrams which are digraphs showing the connections between types of convergence. I don't know if the author invented it , but it seems to be a new way to visualize sequence convergence
processes that I hadn't seen before.
He gets low marks for leaving out Hardy, Hilbert and Banach in his discussion of measure spaces. The place of measure in topology
is also not well discussed. Lebesque measure ideas and integration theories
are well discussed.

Something new in measure theory here
The book isn't organized as a teaching text: is is a definition, theorem and proof type structure that is pretty hard reading. I would have given it two stars except for the convergence diagrams which are digraphs showing the connections between types of convergence. I don't know if the author invented it , but it seems to be a new way to visualize sequence convergence
processes that I hadn't seen before.
He gets low marks for leaving out Hardy, Hilbert and Banach in his discussion of measure spaces. The place of measure in topology
is also not well discussed. Lebesque measure ideas and integration theories
are well discussed. ... Read more

Product Description
Lebesgue Integration on Euclidean Space contains a concrete,intuitive, and patient derivation of Lebesgue measure and integration onRn. Throughout the text, many exercises are incorporated, enablingstudents to apply new ideas immediately.Jones strives to present a slowintroduction to Lebesgue integration by dealing with n-dimensional spacesfrom the outset.In addition, the text provides students a throughtreatment of Fourier analysis, while holistically preparing students tobecome "workers" in real analysis. ... Read more

Customer Reviews (7)

Not For Self Study
[This review has been completely rewritten on October 24, and the ratings have been incremented from 3 stars to 4]

The following review is based on my impression from the first five chapters of the book.

This is a great book for those who are already acquainted with the material. However it is not adequate for self-study for the following three reasons:

1. No official solutions manual is available, either in print or online. This point is particularly painful as the exercises are intimately integrated in the main text, in the sense that results stated in exercises are occasionally referenced in proofs.
2. There is no indication as to the interdependency between the chapters.
3. No official errata sheet is available online.

just enough rigour
This book is truly marvelous. Unlike many other books written by mathematicians, this one does not focus entirely on the overlycomplicated symbolism that has become so frighteningly common in modern textbooks on pure mathematics. The author, for the most part, demonstrates the concept before moving on to nonconstructive methods. This is accomplished via simple pictures, depicting simple figures such as rectangles. By using these to demonstrate the concepts of set theory, and lebesgue measure, the reader almost certainly will develop some awareness of lebesgue integration. For someone not already familiar with set theory (like myself) I suggest writing down the key terms of set theory (presented in the first 23 pages of the book) and keeping them with you while reading. Mathematical rigorousness should come secondary to understanding the key concepts, and therefore the reader might want to skip over nonessential ideas (usually some kind of proof related to an extension of a key concept) and go right to what is needed to understand what is going on.

great!
This is a terrific text for a first course in graduate-level real analysis, and is suitable for self-study.It develops Lebesgue integration theory slowly, in a very clear manner.In addition, the latter part of the book covers the basics of Fourier Analysis and important topics in differentiation.I frequently refer to this book, as the results are easy to find.

Rigor not Rigor Mortis
One of the problems with modern mathematics is its obsession with rigor which has been attended, over the last few decades, by a mushrooming of symbols and jargon.Much of it is not clearly related to the ideas they serve to label, as evidenced by such terms as the topological use of "filter" whose etymology is obscure (ascribed by some to H. Cartan).Moreover, the particular subject of Lebesgue integration and its generalizations is made even more confusing by a wide variety of approaches depending on an author's penchants--many of whom are enamored with a purely axiomatic approach and who make little or no appeal to intuition or--God forbid!--pictures.The author of the present work is obviously someone who has actually taught mathematics and taught it lovingly.This book is an excellent read with lots of interestingtopics well explained from a student's point of view.There seems to be a nice ramping from the truly elementary to the sophisticated, which means the book will interest experienced mathematicians, scientists and engineers.There are lots of "doable" problems that the reader can solve along the way.For the experienced mathematician these little problems help alot as a refresher (Oh!, now I remember, that's how you do it.).I like the emphasis on Euclidean space.Somehow, I always feel more comfortable there!It gives me things I can actually construct and doodle on paper.And, it allows the author to use a few figures in a meaningful way.Which is another of the book's strong points and if I could recommend a future improvement, it would be to bring on more of those pictures!Tristram Needham has done a nice job along these lines with his book "Visual Complex Analysis."(I ordered several copies as Christmas gifts--just kidding!).Anyone who has taught mathematics and genuinely wished to be understood by his students has, at various times, drawn them pictures.Inside the cover sheets are lists of integration formulae, a fourier transform table,and a table of "assorted facts" on things like the Gamma function; which show that this is not only a book on Lebesgue integration but a calculus book with the Lebesgue integral occupying center stage.Everyone who has been enamored by the notion of the integral--as I was as a freshman calculus student and have been ever since--will want to have this book on their shelf.

an excellent introductory text
As someone who wasn't a math major but who has been trying to get up to speed on lebesgue measure and integration, I found this book to be truly accessible. Unlike other "introductory" texts (such as Kopp's "Measure, Integral and Probability") I could follow the reasoning in this book without much difficulty.

The only criticism I have of the book has to do with the first chapter. Its purpose is to provide background mathematical material and given the author's clear ability to explain difficult concepts, I wish that it covered that material in greater detail.

For others who may be looking to build a foundational understanding of this material but who may not be mathematicians, I'd also recommend Pitt's "Measure and Integration for Use" (1985) or his "Integration, Measure and Probability" (1963) (both out of print but fairly easy to find). Those books, along with Jones', are well-used items in my library. ... Read more

Product Description
Designed for use in a two-semester course on abstract analysis, REAL ANALYSIS: An Introduction to the Theory of Real Functions and Integration illuminates the principle topics that constitute real analysis.Self-contained, with coverage of topology, measure theory, and integration, it offers a thorough elaboration of major theorems, notions, and constructions needed not only by mathematics students but also by students of statistics and probability, operations research, physics, and engineering.Structured logically and flexibly through the author's many years of teaching experience, the material is presented in three main sections:Part 1, chapters 1through 3, covers the preliminaries of set theory and the fundamentals of metric spaces and topology. This section can also serves as a text for first courses in topology.Part II, chapter 4 through 7, details the basics of measure and integration and stands independently for use in a separate measure theory course.Part III addresses more advanced topics, including elaborated and abstract versions of measure and integration along with their applications to functional analysis, probability theory, and conventional analysis on the real line. Analysis lies at the core of all mathematical disciplines, and as such, students need and deserve a careful, rigorous presentation of the material. REAL ANALYSIS: An Introduction to the Theory of Real Functions and Integration offers the perfect vehicle for building the foundation students need for more advanced studies. ... Read more

Product Description
This book covers Lebesgue integration and its generalizations from Daniell's point of view, modified by the use of seminorms. Integrating functions rather than measuring sets is posited as the main purpose of measure theory.

From this point of view Lebesgue's integral can be had as a rather straightforward, even simplistic, extension of Riemann's integral; and its aims, definitions, and procedures can be motivated at an elementary level. The notion of measurability, for example, is suggested by Littlewood's observations rather than being conveyed authoritatively through definitions of Sigma-algebras, and good-cut-conditions, the latter of which are hard to justify and thus appear mysterious, even nettlesome, to the beginner. The approach taken provides the additional benefit of cutting the labor in half. The use of seminorms, ubiquitous in modern analysis, speeds things up even further.

The book is intended for the reader who has some experience with proofs, a beginning graduate student for example. It will as well be useful to the advanced mathematician who is confronted with situations -- such as stochastic integration -- where the set-measuring approach to integration does not work. ... Read more

Customer Reviews (1)

at the foundation of mathematics
i found this book in the basement of the math building, i.e. at thefoundation. Get it? pretty clever huh. ... Read more

Product Description
A breakthrough approach to the theory and applications of stochastic integration The theory of stochastic integration has become an intensely studied topic in recent years, owing to its extraordinarily successful application to financial mathematics, stochastic differential equations, and more. This book features a new measure theoretic approach to stochastic integration, opening up the field for researchers in measure and integration theory, functional analysis, probability theory, and stochastic processes. World-famous expert on vector and stochastic integration in Banach spaces Nicolae Dinculeanu compiles and consolidates information from disparate journal articles-including his own results-presenting a comprehensive, up-to-date treatment of the theory in two major parts. He first develops a general integration theory, discussing vector integration with respect to measures with finite semivariation, then applies the theory to stochastic integration in Banach spaces. Vector Integration and Stochastic Integration in Banach Spaces goes far beyond the typical treatment of the scalar case given in other books on the subject. Along with such applications of the vector integration as the Reisz representation theorem and the Stieltjes integral for functions of one or two variables with finite semivariation, it explores the emergence of new classes of summable processes that make applications possible, including square integrable martingales in Hilbert spaces and processes with integrable variation or integrable semivariation in Banach spaces. Numerous references to existing results supplement this exciting, breakthrough work. ... Read more

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This monograph is a comprehensive treatment of the theoretical and computational aspects of numerical integration.
The authors give a unique overview of the topic by bringing into line many recent research results not yet presented coherently; the extensive bibliography lists 268 items. Particular emphasis is given to the potential parallelism of numerical integration problems and to utilizing it by means of dynamic load distribution techniques. The book discusses the basics and provides methodologies for producing efficient and reliable software for numerical integration on advanced computer systems. The book addresses researchers, graduate students, and computational scientists. ... Read more