Product Description
Useful both as a text for students and as a source of reference for the more advanced mathematician, this book presents a unified treatment of that part of measure theory which is most useful for its application in modern analysis. Topics studied include sets and classes, measures and outer measures, measurable functions, integration, general set functions, product spaces, transformations, probability, locally compact spaces, Haar measure and measure and topology in groups. The text is suitable for the beginning graduate student as well as the advanced undergraduate. ... Read more

Customer Reviews (4)

An excellent introduction to measure theory
There comes a time when the budding probabilist or statistician seeks a more comprehensive treatement of measure theory than is afforded in the first few sections of a graduate text in pstats. I chose Halmos's "Measure Theory" for this purpose for two primary reasons: i) Paul Halmos in my opinion is one of the best expository mathematics writers in history, and ii) years ago I paid $1 for the above-mentioned text (original Van Nostrand print) at a local thriftstore. My only perceived drawback was that likely some of his approach to measure theory may be outdated.

After reading the text (up to the chapter on probability) my opinion of Halmos as a writer and mathematician not only has been elevated, but the book delivered the thorough study of measure theory that I had hoped for. Indeed, the author does an excellent job in presenting measure theory in its entire generalitysemi-rings, rings, hereditary rings, algebras, sigma algebras and their extensions are all considered in detail, as well as measures on these set systems: finitely additive , sigma additive, inner measures, outer measures, sigma-finite measures, the completion of measures, regular measures). I especially enjoyed his presentation of Fubini's Theorem along with the concept of "section of a measurable set", which helped the theorem fall out effortlessly. I also found his presentation of different types of convergence (e.g. pointwise, uniform, almost uniform, in measure, in mean) very good and helped give me the bigger picture on modes of convergence. Theorem 22A is essentially a generalization of the Borel-Cantelli Lemma.

The book does have a few downsides. In particular pi and lambda-systems are not used and in some sense replaced by the older notion of semiring. Also, the author's definition of Lesbegue integrable seemed a bit more complicated than what is usually presented (e.g. for nonnegative measurable f to be integrable it requires a sequence fn of simple functions that is mean fundamental and converges in measure to f; compare this with the simpler definition of the integral of measurable f being the sup of Lesbegue integrals of simple functions g for which g <= f). But I consider these downsides minor and highly recommend this text to anyone who seeks a deeper understanding of measure theory.

My impression of measure theory has gone from seeing it asabstract mathematical machinery for simplifying analysis proofs, to a kind of mathematical philosophy that unifies the infinite with the discrete, and lays the proper foundations for inference, probabilistic reasoning, and learning; i.e. the foundations of cognitive intelligence.

Measure Theory -- where are the axioms??
Halmos was a lucid mathematical writer but his "Measure Theory" is dated 1950 and so antedates by two years Rohlin's definitive axiomatic discussion in Doklady Nauk USSR 1952.The attempted axiomatization by Halmos and Von Neumann in the the late 1940's failed and was withdrawn by its authors.Someone who is seriously interested to learn measure theory, with an approach based on the notion of measureable homomorphism, should visit Rohlin's publications and at least one other modern book.
WH Cobbs Narberth PA

A classic in the field
This book is an overview of measure theory that is somewhat dated in terms of the presentation, but could still be read profitably by someone interested in studying the subject with greater generality than more modern texts. Measure theory has abundant applications, and has even gained importance in recent years in such areas as financial engineering. Those interested in the applications of measure theory to financial engineering should choose another book however, since this one does not even mention the word martingale. After a review of elementary topology and set theory in chapter 1, the author begins to define the elementary notions of measure theory in chapter 2. His approach is more general than other texts, since he works over a ring instead of an algebra. Measures on intervals of real numbers is given as an example.Measures and outer measures are defined, and it is shown how a measure induces an outer measure and how an outer measure induces a measure.

The next chapter explores more carefully the relation between measures and outer measures. It is also shown in this chapter to what extent a measure on a ring can be extended to the generated sigma-ring. The all-important Lebesgue measure is developed here also, and the author exhibits an example of a non-measurable set.

In order to develop an integration theory, one must first characterize the collection of measurable functions, and the author does this in chapter 4. The convergence properties of measurable functions are carefully outlined by the author.

The theory of integration begins in chapter 5, wherein the author follows the standard construction of an integral by first defining integrals over simple functions. Then in chapter 6, signed measures are defined, and the Lebesgue bounded convergence theorem is proven and the Hahn and Jordan decompositions of these measures are discussed. The all-important Radon-Nikodym theorem, which gives an integral representation of an absolutely continuous sigma-finite signed measure, is proven in detail.

One can of course take the Cartesian product of two measurable spaces, and the author shows how to define measures on these products in chapter 7, including infinite products. The physicist reader may want to pay attention to the section on infinite dimensional product measures, as it does have applications to functional integration in quantum field theory (although somewhat weakly).

The author treats measurable transformations in chapter 8, but interestingly, the word "ergodic" is never mentioned. He also introduces briefly the L-p spaces, so very important in many areas of mathematics, and proves the Holder and Minkowski inequalities.

The next chapter is the most important in the book, for it covers the notion of probability on measure spaces. After an brief motivation in the first section of the chapter, probability spaces are defined, and Bayes' theorem is discussed as an exercise. Both the weak and strong law of large numbers is proven in detail.

Things get more abstract in chapter 10, which discusses measure theory on locally compact spaces. Borel and Baire sets on these kinds of spaces are defined, and the author gives detailed arguments on what must be changed when doing measure theory in this more general kind of space.

The book ends with a discussion of measure theory on topological groups via the Haar measure. This chapter also has connections to physics, such as in the Faddeev-Popov volume measure over gauge equivalent classes in quantum field theory. The author does a fine job of characterizing the important properties of the Haar measure.

an excellent book
If you want to stydy measure theory from scratch, I do recommend this book. This book is based on a ring, not an algebra, and is a little old-fashioned. So some people feel uncomfortable. But in particular,product spaces, the Fubini theorem and extension theorems are written veryclearly. I'm convinced this book will facilitate your learning in measuretheory and probability theory. ... Read more

Customer Reviews (1)

Wonderful look at mathematics, the times & the author
A Fantastic Book -- this 400+ page manuscript nicely mixes mathematical science with a historical view of the development (1930's through early 1980's) of mathematical research in the United States.This book is highlyreadable, extremely enjoyable and quite straightforward with details andopinions.One gets a first hand insight into how the author approached hisresearch, his career, and his life.Halmos has always been a brilliant andskillful writer but his contributions have mostly been in the technicalarena; this time he has provided a volume we can all enjoy.I found itdifficult to put this book down once I began its reading. ... Read more

Customer Reviews (6)

Text is great, printing is not
This is an absolutely excellent book. I came at it before I had ever taken a linear class. The book develops the subject in a way that it seems a natural progression. I was never faced with a proof that seemed to be just number crunching; they all get at the very heart of what is at stake. I recently read back through it after taking a linear class, and I still come to a better understanding of the subject by looking through this. It is well worth your time. The only complaint I have with the book is that there are a lot of typos. It frequently refers back to the wrong problem number, and there are typos in the mathematics that could cause problems if you aren't keeping track of what is happening. (Ex: in the solution to 135, they forget to specify orthogonal projections, or in 145 they say that if y=Px then what is wanted is Uy=Px when it should actually read Uy=Ax) The text is written clearly enough you should be able to figure out what is happening, but if another edition comes out it would be worth upgrading

Not as good for second year graduate students
It is a good book for beginners but not as great for graduate students.

Great book!
I agree with the following reviewer in that Halmos' books are always entertaining and inviting! Fun to read, concise yet clear! The book contains most of the elementary yet important problems in an undergrad course with solutions in the back. It's very helpful working through all these problem because it will tremendous enhance your understanding of the subject. Also, if you want a hardcore problem oriented approach to linear algebra, check out Proskuryakov's Problems in Linear Algebra. Some of the problems in the book are Putnam like. Virtually any type of Putnam taste problems in Linear Algebra can be found in Proskuryakov.But this one, contrast to Halmos', is the least entertaining--that's why it is called HARDCORE PROBLEM APPROACH! You would be a great linear algebra problem solving machine working through both books!

Start out with one of the best
Halmos is one of the great mathematical expositors of the 20th Century, and his book "Finite Dimensional Vector Spaces" stands as the definitive introduction to the subject for budding mathematicians. This book, "Linear Algebra Problem Book", is perhaps best described as an engaging and semi-informal invitation and complement to that original work, which grew out of lectures given by the legendary John von Neumann. In contrast to typical treatments of linear algebra, "Finite Dimensional Vector Spaces" is abstract (introduces determinants through alternating forms), rigorous, concise, and demands a certain level of mathematical maturity. This book, "Linear Algebra" is exactly the opposite. Starting from very little assumed background, it all but gives away the store, written in plain language, anticipating students' questions and misconceptions, and leading them to a deeper understanding of mathematics through the Socratic method. This is not a problem book in the Schaum's outline sense; there is no drill or rote calculations. Every question is carefully chosen to illustrate a point or expose a potential misunderstanding in the student's knowledge or to exercise the student's intuition and ability to make connections. The answers are given as detailed explanations, integral to the exposition, which go far beyond merely answering the questions posed, raising deeper implications and questions. This is an excellent book for beginning students of higher mathematics, and a very user friendly guide to Halmos' classic text.

Clear, interesting, easy to follow.
I'm no mathematician but, I own a number of linear algebra and abstract algebra books and this book makes them all pale in comparison. I really enjoyed working through it. Maybe a bit pedantic for more advanced typesbut, if you want to get a handle on the concepts this is the book. ... Read more

Product Description

This book is an informal, although systematic presentation of lectures given by the authors on Boolean algebras, intended for advanced undergraduates and beginning graduate students. In a bold and refreshing style, this book treats Boolean algebras, develops some intriguing ideas, and provides rare insights. Exercises are generously sprinkled throughout the text for course study.

This book can be considered a sequel to Paul Halmos's Lectures on Boolean Algebras, with the following changes: (1) the material in every section has been explained in more detail, and is now more accessible to undergraduates; (2) there are three times as many exercises, and the authors have now prepared a solutions manual; (3) a more careful explanation of the relationship between Boolean rings and Boolean algebras has been added; (4) thirteen chapters have been added, including chapters on topology and on continuous functions, a chapter on the extension theorem for homomorphisms, a new chapter on congruences and quotient algebras, a chapter on the lattice of ideals, and a chapter on duality theory for products.

Customer Reviews (2)

Superb introduction. (Contains one flaw.)
As always, Halmos is an excellent expositor. The brief first chapter has been known to scare away physicists and other intelligent people who don't happen to know ring theory, but fortunately the first chapter can be skipped. The book is at the right level to be used by mathematics graduate students learning Boolean algebras and Stone spaces for the first time.

Some people draw a sharp distinction between the concepts of "Boolean space" (a totally disconnected compact Hausdorff space) and "Stone space", the difference being that a Stone space is the Stone space _of_ a Boolean algebra. A Boolean algebra's Stone space is the space of all of its 2-valued homomorphisms with the topology of pointwise convergence of nets of such homomorphisms. That every Boolean space is the Stone space of some Boolean algebra (namely, the Boolean algebra of all of its clopen subsets) is one of the important facts of "Stone's duality". Halmos never mentions the phrase "Stone space", but he proves the basic facts about "Stone's duality": that the category of Boolean algebras and Boolean homomorphisms is the opposite of the category of Boolean spaces and continuous functions.

The flaw is in Lecture 21. Much of that section is founded upon an error -- that a Boolean algebra may have various non-isomorphic completions, of which one is "minimal". A careful mathematician can reconstruct that lecture and get much that is of value, but some may be unfortunately misled.

One other thing irritates me: Halmos uses the word "non-atomic" rather than the much better term "atomless". The problem with "non-atomic" is that it may be mistaken for "not atomic", and that is a quite different thing.

This book is now out of print. I'd like to see Dover reprint it.

ONE OF THE BEST
A very good book, going through the stuff in a very nice kind. Giving you intuitive insights, easy proofs, and written with a large sense of humor. ... Read more

Product Description
Here is an introduction to modern logic that differs from others by treating logic from an algebraic perspective. What this means is that notions and results from logic become much easier to understand when seen from a familiar standpoint of algebra. The presentation, written in the engaging and provocative style that is the hallmark of Paul Halmos, from whose course the book is taken, is aimed at a broad audience, students, teachers and amateurs in mathematics, philosophy, computer science, linguistics and engineering; they all have to get to grips with logic at some stage. All that is needed to understand the book is some basic acquaintance with algebra. ... Read more

Customer Reviews (2)

A superb introduction to the glories of Boolean algebra
This book is an undergrad introduction to Boolean algebraic logic and Halmos, who worked hard in the area during the 1950s, is the person to write it. The book includes Halmos's monadic algebra, but remains at the undergrad level because he stops short of his full-blown polyadic algebra (on which, see Halmos's "Algebraic Logic," which AMS keeps in print and is a fine read).

While Halmos does not cover all of first order logic, he does an excellent job of introducing the reader to the great power and depth of Boolean algebra, revealed by Marshall Stone and Tarski in the 1930s, and other Poles in the 1950s. By this I mean Boolean algebra coupled with the notions of filters, ideals, generators, and quotient algebras. The metatheory of the propositional calculus has a very elegant Boolean representation.

Lattice theory is an extremely powerful generalization of Boolean algebra that has not attracted the attention it deserves. If Halmos had written a text on lattice theory, that situation would in all likelihood have ended. Halmos and Givant include an all-too-brief tantalizing chapter on lattices.

If this book has a drawback, it is the relative unsophistication of its first 40 odd pages, an introduction to logic. This is especially disappointing given that Givant is a logician, and an excellent one at that, being a student of Tarski's.

The books main asset is Halmos's lively prose style, unparalleled in modern mathematics. Math PhD students should study this book closely as a superb example of how to exposit mathematics.

A Builder of a Solid Foundation in Mathematics
It can be strongly argued that logic is the most ancient of all the mathematical sub-disciplines. When mathematics as we know it was being created so many years ago, it was necessary for the concepts of rigidanalytical reasoning to be developed. Of the three earliest areas, geometrywas born out of the necessity of accurately measuring land plots and largebuildings and number theory was required for sophisticated countingtechniques.Logic, the third area, had no "practical" godfather, otherthan being the foundation for rigorous reasoning in the other two. In theintervening years, so many additional areas of mathematics have beendeveloped, with logic and logical reasoning continuing to be thefundamental building block of them all. Therefore, every mathematicianshould have some exposure to logic, with the simple history lessonautomatically being included. This short, but excellent book fills thatniche.
The title accurately sets the theme for the entire book.Algebra is nothing more than a precise notation in combination with arigorous set of rules of behavior.When logic is approached in that way,it becomes much easier to understand and apply. This is especiallynecessary in the modern world where computing is so ubiquitous.Many areasof mathematics are incorporated into the computer science major, but noneis more widely used than logic. Written at a level that can be comprehendedby anyone in either a computer science or mathematics major, it can be usedas a textbook in any course targeted at these audiences.
The topicscovered are standard although the algebraic approach makes it unique. Onesimple chapter subheading, `Language As An Algebra', succinctlydescribesthe theme. Propositional calculus, Boolean algebra, lattices and predicatecalculus are the main areas examined. While the treatment is short, it isthorough, providing all necessary details for a sound foundation in thesubject. While the word "readable" is sometimes overused in describingbooks, it can be used here without hesitation.
Sometimes neglected asan area of study in their curricula, logic is an essential part of allmathematics and computer training, whether formal or informal. The authorsuse a relatively small number of pages to present an extensive amount ofknowledge in an easily understandable way. I strongly recommend thisbook.

Published in Smarandache Notions Journal reprinted withpermission. ... Read more

Product Description

The book is a commemorative volume honoring the mathematician Paul R. Halmos (1916-2006), who contributed passionately to mathematics in manifold ways, among them by basic research, by unparalleled mathematical exposition, by unselfish service to the mathematical community, and, not least, by the inspiration others found in his dedication to that community.Halmos made fundamental contributions in several areas of mathematics.

This volume emphasises Halmos's contributions to operator theory, his venue for most of his mathematical life. The core of the volume is a series of expository articles by prominent operator theorists providing an overview of how operator theory prospered during the Halmos era, in no small measure thanks to Halmos's leadership and penetrating insights.

Product Description
This classic book is based on lectures given by the author at the University of Chicago in 1956. The topics covered include, in particular, recurrence, the ergodic theorems, and a general discussion of ergodicity and mixing properties. There is also a general discussion of the relation between conjugacy and equivalence. With minimal prerequisites of some analysis and measure theory, this work can be used for a one-semester course in ergodic theory or for self-study.ReadershipGraduate students and research mathematicians interested in number theory.Table of ContentsIntroduction Examples Recurrence Mean convergence Pointwise convergence Comments on the ergodic theorem Ergodicity Consequences of ergodicity Mixing Measure algebras Discrete spectrum Automorphisms of compact groups Generalized proper values Weak topology Weak approximation Uniform topology Uniform approximation Category Invariant measures Invariant measures: the solution Invariant measures: the problem Generalized ergodic theorems Unsolved problems References ... Read more

Customer Reviews (1)

A Clear Exposition of Ergodic Theory
I am a retired mathematician and was curious as to what ergodic theory was about.I bought several books on the subject and found Halmos' book by far the the most helpful.He leads one into the subject in a straightforward manner avoiding the use of a tiresome complicated system of terminology and symbols.He only assumes a knowledge of analysis in the reader and in other parts of mathematics either enunciates the result to be used or gives a sufficient reference to it.His exposition is succinct and well phrased, and he gives many examples to illustrate the conceptions he introduces. ... Read more

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Customer Reviews (15)

I got the book itself in great condition, BUT
Let's get one thing straight...if you're going to buy this book, you need to be prepared to teach yourself some of the material on your own. It's a very compact text, and they cover a lot of material in a matter of a few pages. The author isn't going to hold your hand with lengthy examples and proofs with plenty of space in between paragraphs. In short, this isn't a linear algebra text for pretenders. It's serious, and it takes dedication to work with. If you want a book that takes more time to explain concepts instead of assuming you will get it the first time, I recommend something like "Linear Algebra Done Right".

Amazing.
Even though I am not an undergraduate student (yet), I have to point out that this book is amazing as a first read for one good reason: Halmos forces the reader to prove some nontrivial results that are generally proved in other textbooks.I particularly like the fact that the author refers to infinite dimensional vector spaces in the first few chapters.The famous saying among problem solvers is that "problem solving can only be learnt through solving problems."Due to the fact that a mathematician needs to know how to approach problems on his or her own, he or she must be FORCED to solve problems.Working through this book is worth the time.However, some experience with proofs is a must; other that this, the book is self-contained for those who are familiar with some real analysis.

Linear Algebra Bible
This book is simply awesome; it involves just the right amount of (fundamental) concepts and ideas in linear algebra, presented with the utmost rigor by a master logician.

Good Reading.
Halmos uses informal language and is not afraid to let his feelings show from time to time. There are a couple of exasperated remarks about the formalism that I find very funny. Because of this personal quality of the prose I feel affection for the work that I don't feel for many textbooks.

For me this book was just fine as the introductory text. But I was a physics student and must have had from that a working knowledge of vectors and transforms in orthogonal spaces, and I knew and liked the rigorous approach to math.

I still feel I would not like to be without the book.

A Classic for the mathematically-inclined.Good preparation for learning quantum mechanics.
This was one of the two textbooks (along with Rudin's Principles of Mathematical Analysis) that was used for the hot-shot freshman Math 218x course taught by Elias Stein at Princeton some years ago.

It is a great book, one of my all-time favorites.It requires a bit of mathematical maturity, that is a love of mathematical proof and simplifying abstractions.This book abstractly defines vector spaces and linear transformations between them without immediately introducing coordinates.This approach is vastly superior to immediately extorting the reader to study the algebraic and arithmetic properties n-tuples of numbers (vectors) and matrices (n x n tables of numbers) which parameterize the underlying abstract vectors and linear transformations, respectively.

If I taught a serious linear algebra course using this book then there are a few deficiencies I would try to correct:

1. The polar decomposition is covered but the singular value decomposition (for linear transformations between different inner product spaces) is omitted.This is a pretty big gap in terms of applications, although it's easy to get the singular value decomposition if you have the polar decomposition.

2. The identification of an reflexive vector space with its double-dual was a stumbling block for me when I took the course.There was no mathematical definition of "identify", and so I was confused.Perhaps a good way to remedy this is to give a problem with the example of the Banach space L^p (perhaps just on a finite set of just two elements), and show how L^p is dual to L^p'.

3. The section on tensor products should be improved and expanded, especially in light of the new field of quantum information theory.

4. It would be nice to have a problem (or take-home final) where the reader proves the spectral theorem using minimal polynomials without recourse to determinants, and introduces the functional calculus just using polynomials.It is disturbing to see how many physics grad students are so hung up thinking of eigenvalues only as roots of the characteristic polynomial that they can't understand properties of the spectrum of a self-adjoint transformation A by considering polynmomials of A.

5. I missed the connection between polynomials of a matrix and the Jordan form when I learned linear algebra from this book.Perhaps the following problems would be helpful, and give a proper finite-dimensional introduction to the Dunford calculus (before it is slightly-obfuscated in infinite dimensions using Cauchy's formula):

Problem A: Let P be a complex polynomial, and let A be a linear transformation on a complex vector space, with eigenvalues {z_1,...,z_n}, and let the Jordan block corresponding to z_k have a string of 1's that is at most s_k elements long.Then the value of P(A) is determined by the values of P and its first s_k derivatives at the z_k.(One defines the derivative of a function from C to C by taking a limit of difference quotients, in the same way one defines a derivative of a real function.In particular, the usual rules for differentiating polynomials apply.)

Problem B: (Finite-dimensional Dunford calculus, assuming differentiablity only on the spectrum) Suppose that f:C->C has s_k complex derivatives at the z_k. Define f(A)=P(A), where P is a polynomial with derivatives up to order s_k agreeing with those of f at the z_k.Show that such polynomials always exist.(In particular, f(A) is well-defined by problem A.) Show that (f+g)(A)=f(A)+g(A), f(A)g(A)=(fg)(A), and f(g(A))=h(A), where h is the composition of f and g as functions from C->C.

Problem C: Use B to show that every nonsigular matrix has a square root, as do singular matrices with no 1's in the jordan block for the eigenvalue 0.

Problem D: Are the only matrices with square roots given by problem C?

Except for property (3) above, this is a good book for students who are interested in taking a quantum mechanics or quantum computing course in the future.

If you read this book and like it, then in the future you might want the following graduate-level textbooks:

Bhatia's book "Matrix Analysis".

Reed and Simon's "Methods of Mathematical Physics", especially volume 1 on functional analysis.(This is the infinite-dimensional version of Halmos's book.)

Halmos's "A Hilbert Space Problem Book"

You'll certainly need to learn some analysis before tackling the last two books, though! ... Read more

Product Description
Paul R. Halmos, eminent mathematician and brilliant expositor, is also a snapshot addict. Between 1943 and 1988, Halmos photographed mathematicians, their spouses, their brothers and sisters and other relatives, their offices, their dogs, and their carillon towers. From the 6000 or so photographs in his collection, Halmos has chosen about 600 for this book. The pictures are candid shots showing mathematicians just being themselves. The accompanying captions, in addition to identifying the subjects, contain anecdotes and bits of history that reveal Halmos's inimitable wit, charm, and insight. This delightful collection of mathematical memorabilia has become a favorite browsing book as well as a valuable historical record. It is now available in a paperback edition. ... Read more

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Routledge is now re-issuing this prestigious series of 204 volumes originally published between 1910 and 1965. The titles include works by key figures such asC.G. Jung, Sigmund Freud, Jean Piaget, Otto Rank, James Hillman, Erich Fromm, Karen Horney and Susan Isaacs. Each volume is available on its own, as part of a themed mini-set, or as part of a specially-priced 204-volume set. A brochure listing each title in the International Library of Psychology series is available upon request. ... Read more

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Originally published in 1952.