Customer Reviews

The Elements of Integration and Lebesgue Measure

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When I took my first one-semester course on measure and Lebesgue integration my teacher chose Bartle's "The Elements of Integration" as text. After reading many other books on the subject now I'm sure he made a wise decision.

Assuming almost no strong mathematical background, Bartle is able to build up the basic Lebesgue integral theory introducing the fundamental abstract concepts (sigma-algebra, measurable function, measure space, "almost everywhere", step function, etc.) in such an easy way that the student is not only able to handle them but to UNDERSTAND them.

From the first part of the book I appreciate specially chapters 6, 7, and 10, on L_p spaces, modes of convergence, and product measures, respectively. These chapters contain the most used results of the basic theory, and they are stated exactly in the way one needs them, making the book very useful for future reference.

I like the second part very much also, because it stresses the importance of measure theory by itself and not only as a requisite for integration theory. If you are interested in fractal geometry or geometric measure theory you will find chapters 11 to 17 very helpful.

Since I own this book it has never been lazy in my bookshelf.

Measure and Integration is a daunting subject for mathematical neophytes. Bartle's little volume is the right place to start. I first learned measure theory from it 20 years ago and went on to study functional analysis and stochastic approximation.

I was able to master the material on my own with this book. The problems are at the right level and he begins with the correct level of abstraction. I recommend it over anything else because it is straighforward, clear and focused. Master it then go on to Walter Rudin's Real and Complex Analysis.

This book deals with integration theory in an abstract level (measure theory). It's "straight to point" and should be a reference for senior undergarduates and graduates; well written and wisely structured. I'm a senior undergraduate in statistics and studied this book in a summer course as a preparation for a following measure-theorethic probability course! An excellent pre-requisite for Prof. Bartle's book is his other text 'Elementary Real Analysis', which is also a great textbook. Those interested in speciliasing in measure theory may also check Wheeden and Zygmund: 'Measure and Integration' and Lang: 'Real and Functional Analysis'. For the serious statistician and the mathematician I strongly recommend Bartle's two books, Kolmogorov and Fomim: 'Introductory Real Analysis' (more than an excellent book, Kolmogorov's text is actually an introduction to topology and measure) and Billingsley: 'Probability and Measure'; these books do please the reader and not (only) the authors!!!

The exposition of integration in this book is the clearest I have read. I also found the chapter on modes of convergence, where it laid out the relationship between things such as L^P-convergence and convergence in measure, to be extremely useful. The second half, where it covers topics like Lebesgue measure, repeats some of the same information from the first part which is a bit iritating if you are reading straight throught, but contains a lot of good information. The book is also quite small making it easy to take with you as a quick reference.

Let me warn you though that this is an introduction to integration and measure _not_ an introduction to real analysis. It does not cover important topics like L^P-approximation, differentiation, etc. For a complete treatment of real analysis, I recommend the books "Lebesgue Integration on Euclidean Space" by Frank Jones and the slightly more abstract "Real and Functional Analysis" by Serge Lange.

IF YOU WANT TO UNDERSTAND MEASURE THEORY READ THIS BOOK, MAYBE THE ONLY PROBLEM IS THE LACK OF EXAMPLES BUT THE WAY THAT THE THEORY IS PRESENTED MAKE IT YOUR FIRST CHOICE WHEN YOU TRY TO LEARN MEASURE THEORY.

The book is concise and easy to follow. The author rarely gives lengthy explanations and analogies, but spends the bulk of the book stating solid facts and proofs. I also like the organization of the book. All definitions and theorems are explicitly stated and indexed, not scattered in paragraphs in the body of the text.

The book misses subjects such as complex measures (they are briefly mentioned), the fundamental theorem of calculus under Lebesgue settings, and probability measures, but its ok since the book is an introduction to the subject. A more comprehensive (and harder to read) book is "Real & Complex Analysis" by Walter Rudin. If you are interested in probability, consider Ptrick Billingsley's book "Probability and Measure".

This is a great introduction to integration theory, and probably the only book that is suitable for (1) a one semester course, and (2) a wide audience. The topics presented in this book are the main concepts ("work horse material") of integration and measure. I highly recommend this book as an introduction or tag along book for mathematicians (analysts), and as a primary book to learn the material for students in economics, electrical engineering, and statistics.