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Byathan_tolis@yahoo.comon 7 April 2002

I bought this book because I was enrolled in a Stochastic Processes class and was looking for a good, easy self-study book to understand Lebesgue integration and probability. I am extremely pleased I bought the book and actually spent three weeks of my life working out every single problem. The good news is that I came out with a solid understanding of real analysis, and integration in particular. The bad news is that probability is given short shrift, a very small proportion of the worked examples and an even smaller proportion of correct answers to the worked problems. With that said, the treatment of real analysis is so accessible and so thorough that I am still very happy with the book. It brought back memories of my freshman year in college.

The book is made up of seven chapters: 1. Intro 2. Measure 3. Measurable functions 4. Integral 5. Elements of (relevant) functional analysis 6. Product measures 7. Modes of Convergence, Strong and Weak laws and CLT. The seventh chapter is 100% probability. The first six are not. They first do a fantastic job of explaining the real-analysis concepts and then, almost as an afterthought, have a few tacked-on paragraphs that explain how it all translates in probability terms. For example, in the Measure chapter we are told what a probability measure is and in the Integration chapter the authors introduce expectation, but their heart really isn't in it. Either that or it's a case of good author and bad author.

Regardless, this is a book I can wholeheartedly recommend, because the 75% of the book that does not regard probability is a true five-star job.

A few final comments:

1. It is a true, honest-to-God self-study guide that a semi-awake undergrad can follow. Have no fear.

2. Contrary to the description, I did not learn Radon-Nikodym from here.

3. Comfortably the worst appendix of any book I have ever bought.

4. Loads of errata, as it's a first edition. Here's a few I think I got:

p.39 "i" should range from zero to n, not infinity in the last summation

p.106 {x:<y} should read {x<y} (I'm nitpicking)

p.113 the proof of proposition 4.5 is incomplete

p.126 need to mention (a,zb) = conjugate of z(a,b) for the brave readers who will attempt to prove the polarization identity

p.149 Aù2 is not in F2

p. 150 and 152 integration over omega 1 and omega 2 is consistently backwards, undermining the entire discussion. In other words, when he is integrating over omega 1 he ought to be integrating over omega 2 and vice versa.

p. 151 (THIS COST ME AN HOUR OF MY LIFE) in the last three lines A1, A2, Ai should read B1, B2, Bi and all Bs should be As.

p. 208 "This implies convergence to zero of {funky expression} almost surely." I disagree.

5. Even more errata in the solutions to the exercises. Definitely done by grad students. I disagree with the answers to 4.1.b. 5.3, 5.4. a, b and c, 6.5, 6.6 (only half the answer), 6.8 and 7.4

With all that said, this is a superb book, unless you are buying it to learn probability theory!

The book is made up of seven chapters: 1. Intro 2. Measure 3. Measurable functions 4. Integral 5. Elements of (relevant) functional analysis 6. Product measures 7. Modes of Convergence, Strong and Weak laws and CLT. The seventh chapter is 100% probability. The first six are not. They first do a fantastic job of explaining the real-analysis concepts and then, almost as an afterthought, have a few tacked-on paragraphs that explain how it all translates in probability terms. For example, in the Measure chapter we are told what a probability measure is and in the Integration chapter the authors introduce expectation, but their heart really isn't in it. Either that or it's a case of good author and bad author.

Regardless, this is a book I can wholeheartedly recommend, because the 75% of the book that does not regard probability is a true five-star job.

A few final comments:

1. It is a true, honest-to-God self-study guide that a semi-awake undergrad can follow. Have no fear.

2. Contrary to the description, I did not learn Radon-Nikodym from here.

3. Comfortably the worst appendix of any book I have ever bought.

4. Loads of errata, as it's a first edition. Here's a few I think I got:

p.39 "i" should range from zero to n, not infinity in the last summation

p.106 {x:<y} should read {x<y} (I'm nitpicking)

p.113 the proof of proposition 4.5 is incomplete

p.126 need to mention (a,zb) = conjugate of z(a,b) for the brave readers who will attempt to prove the polarization identity

p.149 Aù2 is not in F2

p. 150 and 152 integration over omega 1 and omega 2 is consistently backwards, undermining the entire discussion. In other words, when he is integrating over omega 1 he ought to be integrating over omega 2 and vice versa.

p. 151 (THIS COST ME AN HOUR OF MY LIFE) in the last three lines A1, A2, Ai should read B1, B2, Bi and all Bs should be As.

p. 208 "This implies convergence to zero of {funky expression} almost surely." I disagree.

5. Even more errata in the solutions to the exercises. Definitely done by grad students. I disagree with the answers to 4.1.b. 5.3, 5.4. a, b and c, 6.5, 6.6 (only half the answer), 6.8 and 7.4

With all that said, this is a superb book, unless you are buying it to learn probability theory!

ByGiacomo Bruzzoon 20 September 2004

Having studied extensively through the first edition of this book, I was very aware of its qualities and defects. It was one of the first of its kind (the only other Introductory Measure&Integration monograph I know of providing answers to exercises for the lonely climber is Ash and DoleansDade - a very very fine book indeed), it was set in readable print (although some people cringe when looking at latex..) and most importantly it was very very well written. Unfortunately it was peppered with a number of typos, some quite irritating. But time passed, the typos were corrected, and we are now provided with a new edition which (FINALLY!) contains material on the Radon-Nikodym theorem and Lebesgue Stieltjes measures. Most importantly the links between existence of density, absolute continuity, RN derivative are all clearly established (the irritating - please forgive me - Skorohod argument has left its place to the intuitive construction of the Lebesgue Stieltjes measure - thus simplifying the transition from measure probability theory, but also risk-neutral pricing (pricing kernels and all that)).

The inclusion of material on the Hahn-Jordan, Riesz and Doob-Meyer decompositions (forgive the order) only makes this book more desirable and interesting.

Many, Many thanks to Professors Capinski and Kopp for pushing these changes through.

One last issue: Dear Prof Kopp, please have your book on Martingales and Stochastic Calculus reprinted....

The inclusion of material on the Hahn-Jordan, Riesz and Doob-Meyer decompositions (forgive the order) only makes this book more desirable and interesting.

Many, Many thanks to Professors Capinski and Kopp for pushing these changes through.

One last issue: Dear Prof Kopp, please have your book on Martingales and Stochastic Calculus reprinted....

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ByNTon 13 August 2007

I've dredged through quite a few other books and internet material on this subject. But have always been left short of the big picture and motivation behind the basic ideas.

Most other presentations just lay down the theorems, fine if your a maths whizz, not so good if your struggling through your own course of self-study.

This book has useful remarks and comments before/after various definitions and theorems - so you don't feel like they're just being pulled out of the hat. These remarks were of inestimable value and kept providing 'AHA! that now makes sense' moments.

If you've ever struggled in this area of mathematics, I would thoroughly recommend you give this a try.

Most other presentations just lay down the theorems, fine if your a maths whizz, not so good if your struggling through your own course of self-study.

This book has useful remarks and comments before/after various definitions and theorems - so you don't feel like they're just being pulled out of the hat. These remarks were of inestimable value and kept providing 'AHA! that now makes sense' moments.

If you've ever struggled in this area of mathematics, I would thoroughly recommend you give this a try.

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ByA customeron 18 February 2000

Many years ago I took a course in Real Analysis using the standard text by Royden. Though I labored mightily, much of the material did not make sense to me. Subsequent self-tuition finally enabled me to master the material, but with an enormous investment of time.

If only "Measure, Integral and Probability" had been available to me at that time! This book very clearly and simply introduces the basic concepts of measure and integral, in a way that will greatly benefit the person intending to make a formal study Real Analysis. The exercises are very carefully chosen, and the solutions in the back of the book are accessible.

This book will also be useful to the person who just wants some idea of what measure theory is all about, and has no intention of pursuing Real Analysis. Though why any person would not want to learn Real Analysis is beyond me. :)

At the time I studied Real Analysis, I searched long and hard for just such a book as this, and found none. I doubt that any other has been published. Capinski and Kopp are to be commended for their ability to bring measure theory to the mere mortal.

- an anonymous person who is embarrassed to admit that he had difficulty mastering Royden's text

If only "Measure, Integral and Probability" had been available to me at that time! This book very clearly and simply introduces the basic concepts of measure and integral, in a way that will greatly benefit the person intending to make a formal study Real Analysis. The exercises are very carefully chosen, and the solutions in the back of the book are accessible.

This book will also be useful to the person who just wants some idea of what measure theory is all about, and has no intention of pursuing Real Analysis. Though why any person would not want to learn Real Analysis is beyond me. :)

At the time I studied Real Analysis, I searched long and hard for just such a book as this, and found none. I doubt that any other has been published. Capinski and Kopp are to be commended for their ability to bring measure theory to the mere mortal.

- an anonymous person who is embarrassed to admit that he had difficulty mastering Royden's text

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ByGekaon 14 April 2009

I wish I had this book when I was enrolled in the course on financial math. As someone with non-math background and being used only to calculus based proba I had loads of trouble with the introduction to measure theory. Capinski and Kopp do a really great job making things clear! You get loads of intuition and rigor in the same pack! If you want to study measure and proba on your own this book is a great place to start!

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ByMetrogeekon 24 July 2012

I cannot offer much new commentary but I thought I'd get the book's 5-star count up.

It is an excellent bridge from analysis (via Royden or Rudin) to probability and *does* fill a gap in the literature. It features very doable exercises and propositions for those inclined to self-study. With a coherent narrative, it pieces together all the scattered miscellany of foundational analysis that no author (to my knowledge) has cared to explain in one place (at least in financial maths).

The "voice" of the text is very welcoming which again offers relief from the far too many staid treatments of similar topics.

Well done, sirs.

It is an excellent bridge from analysis (via Royden or Rudin) to probability and *does* fill a gap in the literature. It features very doable exercises and propositions for those inclined to self-study. With a coherent narrative, it pieces together all the scattered miscellany of foundational analysis that no author (to my knowledge) has cared to explain in one place (at least in financial maths).

The "voice" of the text is very welcoming which again offers relief from the far too many staid treatments of similar topics.

Well done, sirs.

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ByThe Green Flagon 11 August 2011

Many undergraduate mathematics or mathematics and statistics courses cover probability to quite an advanced level but do not cover the required underlying formal mathematics or do so cursorily. Unless you are able to choose a course in measure theory you may be left with a gap in your knowledge, especially if you later wish to study more advanced topics in detail.

If you have a good understanding of real analysis then this book will give you a straightforward introduction to measure theory and the theory of (Lebesgue) integration up to the Radon-Nikodym Theorem with additional applications to probability theory (including (discrete) martingales) (and some financial mathematics). You won't learn probability theory from this book in any applied sense but that is not the aim here. In fact you are really expected to have some knowledge of probability theory.

I found the book to be well laid out for its purpose and very easy to follow for self-study. There may be better books out there with similar aims but this one is more than adequate.

If you have a good understanding of real analysis then this book will give you a straightforward introduction to measure theory and the theory of (Lebesgue) integration up to the Radon-Nikodym Theorem with additional applications to probability theory (including (discrete) martingales) (and some financial mathematics). You won't learn probability theory from this book in any applied sense but that is not the aim here. In fact you are really expected to have some knowledge of probability theory.

I found the book to be well laid out for its purpose and very easy to follow for self-study. There may be better books out there with similar aims but this one is more than adequate.

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ByMr. G. Zhouon 30 January 2011

This book helps your University statistics knowledge, good research book for final year projects and... Also a good book to have in the open book tests XD

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