Jürgen Elstrodt - Maß- und Integrationstheorie (only in German). Fremlin - Measure Theory (freely available in the web space, contains pretty. Donald L. Cohn-"Measure theory". Everything is In addition, you need a good book on Lebesgue measure on Euclidean spaaces. For this I. This is a preliminary version of the book An Introduction to Measure Theory published by to measure theory on Euclidean spaces Rd (starting with the more.
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The 44 best measure theory books recommended by Nassim Nicholas Taleb, such as Real Analysis, Ergodic Theory and Probability Theory. A list of 4 new measure theory books you should read in , such as A Concise Introduction to Measure Theory. From the book reviews: “This textbook provides a comprehensive and consistent introduction to measure and integration theory. The book can be.
Terry Tao has notes about the subject on his blog, see here. One of the most comprehensive books, besides Kallenberg's Foundations of Modern Probability, is probably Bogachev's Measure Theory 2-volumes. Its Table of Contents can be viewed at Springer.
Chapters 1, 2, 3, 6, 7 and 8 constitute an excellent general treatment of measure theory. Let me elaborate:. Chapter 1: The notions of an abtract measure space and an abstract topological space are introduced and studied in concurrence.
This treatment allows the reader to see the close connections between the two subjects that appear both in practice and in theory. Elementary examples and properties of measurable functions and measures are discussed. Furthermore, Lebesgue's monotone convergence theorem, Fatou's lemma, and Lebesgue's dominated convergence theorem are proven in this chapter. Finally, the chapter discusses consequences of these results.
The elegance of the treatment allows the reader to quickly become accustomed to the basic theory of measure. Chapter 2: This chapter delves further into the intimate connection between topological and measure theoretic notions. More specifically, the chapter begins with a treatment of some important results in general topology such as Urysohn's lemma and the construction of partitions of unity.
Afterwards, these results are applied to establish the Riesz representation theorem for positive linear functionals. The proof of this result is long but is nonetheless carefully broken into small steps and the reader should find little or no difficulty in understanding each of these steps. The Riesz representation theorem is applied in a particularly elegant manner to the theory of positive Borel measures.
Finally, the existence and basic properties of the Lebesgue measure are shown to be a virtually trivial consequence of the Riesz representation theorem. The chapter ends with a nice set of exercises that discusses, in particular, some interesting counterexamples in measure theory.
Chapter 3: The chapter begins with an elementary treatment of convex functions. Rudin explains that many elementary inequalities in analysis may be established as easy consequences of the theory of convex functions and evidence is provided for this claim. In particular, Holder's and Minkowski's inequalities are proven.
Chapter 6: This chapter discusses the theory of complex measures, and in particular, the Radon-Nikodym theorem. These results culminate in the proof of the Riesz representation theorem for bounded linear functions. Uniform integrability and the Vitali convergence theorem are treated in the exercises at the end of the chapter.
Chapter 7: The main topic of this chapter is Fubini's theorem. A wealth of nice counterexamples is discussed and an important application is presented: A wonderful feature of this treatment is the generality; the result is established in one of the most general forms possible. Chapter 8: This chapter treats differentiation of measures and the Hardy-Littlewood maximal function which is an important tool in analysis.
Several results from this chapter are also used later in this book; most notable is the use of the differentiation theorem of measures in the study of of harmonic functions in chapter A good knowledge of set-theoretic notions, continuity and compactness suffice for the chapters that I have described. An at least rudimentary knowledge of differentiation and uniform convergence is very helpful at times. One need not be acquianted with the theory of the Riemann integral beforehand although one should at least be acquianted with its computation.
In short, a knowledge of chapters 1, 2, 3, 4 and 7 of Rudin's earlier book Principles of Mathematical Analysis is advisable before one reads this textbook.
The exercises in this textbook are wonderful. Many of the exercises build an intuition of the theory and applications treated in the text and therefore it is advisable to do as many exercises as possible.
However, you should expect to work to solve a few of the exercises.
However, if you are truly stuck you will find that many of these results are either theorems or exercises with detailed hints in other textbooks. I have already described the content in some detail but let me say that the content is about exactly what one needs to study branches of mathematics where measure theory is applied. Of course, this is with the assumption that one at least attempts as many exercises as possible since a number of important results from probability theory, for example are treated in the exercises.
The proofs in Rudin are with possibly minor exceptions complete.
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Unlike a number of other mathematics textbooks, Rudin prefers not to leave any parts of proofs to the reader and instead focusses on giving the reader non-trivial exercises as practice at the end of each chapter. The book reads magnificently and the flow of results is excellent; almost all results are stated in context.
It is fair to say that the main text of the book lacks examples, which is perhaps one of the only points of complaints by students, but the exercises do contain examples. Finally, the book is rigorous and is completely free of mathematical errors. I hope this review of Rudin's Real and Complex Analysis is helpful! It really motivated me to delve deeper into analysis. Perhaps the same will be true for you. I certainly recommend this book with my deepest enthusiasm.
Folland's text "Real Analysis" is highly extensive and covers many topics in measure theory which you rarely see in other books, e.
It also has a chapter on probability theory, in which he gives rigorous proofs to the basic theorems in the theory the law of large numbers, the central limit theorem , talks about the construction of product spaces in the context of probability theory, and discusses Brownian motion and Wiener measure.
When I learned the subject, I found three books to be immensely useful. Royden's Real Analysis is a good general book and has nice problems. Bartle's elements of integration does the abstract theory of integration cleanly and concisely.
In addition, you need a good book on Lebesgue measure on Euclidean spaaces. It seems unnecessary to add to this long list of great books, but Real Analysis and Probability by R. Dudley is wonderful. His book fits your need to emphasize on both aspects of analysis and probability.
If you are looking for a book in measure theory, you should certainly get a copy of the book of that title by Halmos. You may need a second book for details on stochastic processes, but for the underlying analysis it will be hard to find a more comprehensive book, or a better-regarded author. If you want a book to be a comprehensive study of measure theory, you can hardly be more extensive than the five volumes by Fremlin.
Measure Theory and Integration
Perhaps my answers will be idiosyncratic, since I'm a philosopher of science who hasn't taken any math since Calculus apart from some logic courses, and my main interest in measure theory was for the sake of probability theory. David Williams, Probability with Martingales is great for concepts as well as proofs, but there's a lot that it doesn't cover, I feel.
I've gotten a lot out of J. Doob's Measure Theory , which presents some common ideas in ways that are more general and deeper than what one usually finds, I believe.
Has a bit of droll humor, now and then, too. I don't understand all of it, but I've gotten a lot out of it.
This was where I started. Munroe's Introduction to Measure and Integration fits well with Friedman's book. I used them together. OK, and now feel free to ignore this. Most of the other answers are probably by people who are better informed.
Nobody seems to mention the book "Measure and Integration" by De Barra. It covers all the standard topics and is very detailed.
The exercises have detailed solutions too. Richard F. In my opinion, his treatment of integration is the best one I have ever seen. And then he defines it for complex valued functions. On the other hand, Lang defines it for real or complex valued functions all at once. And his method also applies without any modification to functions taking values in a Banach space.
Most of all, his method is simple, clear and natural. Mathematicians from other areas as well as physicists, computer scientists, engineers and econometrists will find useful results and powerful methods for their research. The reader may find in the Handbook many close relations to other mathematical areas: real analysis, probability theory, statistics, ergodic theory, functional analysis, potential theory, topology, set theory, geometry, differential equations, optimization, variational analysis, decision making and others.
The Handbook is a rich source of relevant references to articles, books and lecture notes and it contains for the reader's convenience an extensive subject and author index.
Some elements of the classical measure theory E. Paradoxes in measure theory M. Convergence theorems for set functions P. Differentiation B. One-dimensional diffusions and their convergence in distribution J. The Riesz Theorem J. Diestel, J. Stochastic processes and stochastic integration in Banach spaces J. Part 3, Integration theory Daniell integral and related topics M. Pettis integral K. The Henstock-Kurzweil integral B. Integration of multivalued functions Ch. Part 4, Topological aspects of measure theory Density topologies W.
FN-topologies and group-valued measures H. On products of topological measure spaces S. Perfect measures and related topics D.
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Part 5, Order and measure theory Riesz spaces and ideals of measurable functions M. Measures on Quantum Structures A. Probability on MV-algebras D. Mundici, B. Measures on clans and on MV-algebras G. Barbieri, H. Triangular norm-based measures D.Stein, E. That said, it doesn't say much about measures other the one referenced in the title presumably to keep the prerequisites minimal.
That kind of question is hard to vote on. I should mention that the problems are accessible to students—some of the exercises are especially recommended for students ….
Elementary examples and properties of measurable functions and measures are discussed. It includes a lot of classical material that many books ignore. Royden's Real Analysis is a good general book and has nice problems. The chapter ends with a nice set of exercises that discusses, in particular, some interesting counterexamples in measure theory. Folland, G. Post as a guest Name.
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