# Definition problem of the integral of a nonnegative f in royden's book

1. Jul 21, 2009

### alexzhu

Hi guys,

Does anyone notice a definition problem in royden's book?

In chapter 4, section 3, royden's definition: the integral of a nonnegative measurable function f over a measurable set E to be the supremum of all the integrals of bounded measurable functions (Each of them vanishes outside a set of finite measure and is no greater than f) over E.

Here is the problem: assume the measure of E is infinite, then an integral of a bounded measurable function (vanishes outside a set of finite measure and no greater than f) over E is not defined, because in section 2 royden only defines an integral of a bounded measurable function over a set with finite measure. So there seems to be a problem in royden's definition above. Strictly speaking, integrals of bounded measurable functions over E are not defined before taking supremum.

If you have a background of real analysis before reading royden's book, then this definition problem could be solved by other means. However, if you are a beginner and your knowledge of real analysis is limited to the materials from chapter 1 to chapter 4, section 2 in royden's book, then this definition problem seems to be unsolvable.

Hope I make my point clear, thanks for any help!

2. Jul 29, 2009

### n!kofeyn

It seems you are right, although I don't find it surprising. It still surprises me this book is so popular.

3. Jul 30, 2009

### alexzhu

I guess the reason for its popularity is that it provides sort of "classic" viewpoints to deal with real analysis, which is of course different from the modern techniques. Anyway, it is an excellent book as a reference.

4. Jul 30, 2009

### n!kofeyn

This book is horrible as a reference! I really can't stand the book. It hardly offers a viewpoint, as it is a collection of theorems, propositions, proofs, and exercises. There are no insightful discussions and the definitions are buried within paragraphis, which doesn't make for a quick look at what a definition actually says. The index is horrible. As an example, a major portion of this book is over Lebesgue something. There is Lebesgue measure, the Lebesgue integral, Lebesgue-Stieltjes integral, Lebesgue monotone convergence, Lebesgue bounded convergence, Lebesge dominated convergence. Are these in the index? No. Lebesgue decomposition is the only entry in the index under Lebesgue.

He doesn't even dare mention Henri Lebesgue and his goals or descriptive definition of the Lebesgue integral. He makes no mention that there are other methods of defining measurable sets other than the Caratheodory condition, as he merely just states it. I find it to be a sad testimony to the fact that professors aren't more innovative when choosing a textbook.

5. Jul 31, 2009

### eof

The best book I know is Folland's Real Analysis. Although the proofs are terse you rarely have to spend much time filling in the details. It's strengths are that it has very good discussions after each chapter. Unfortunately, the book has some typos, but the author maintains a very good errata list.

6. Jul 31, 2009

### alexzhu

I couldn't agree more! I'm reading it. Seemed that Wheeden and Zygmund's book is also popular in the math community.

Last edited: Jul 31, 2009
7. Jul 31, 2009

### alexzhu

I don't notice that until you point it out. Some of the definitions in the book are indeed annoying and ambiguous, "simple function" for example. I really don't like them. I stopped reading after going through 4 chapters.