Replacement for Rudin Chapters 9-11

[SrVishi]

I plan on working through Rudin's Principles of Mathematical Analysis until chapter 8 since I hear 1-8 are fantastic (which I agree with right now), but 9-11 are abysmal. Does anyone know of any good book from which I could replace these chapters? I believe they cover multivariate/vector analysis, differential forms, and Lebesgue integration. Thanks in advance for any response.

 

[Geofleur]

I'm not sure if there is a single real analysis book besides Rudin's that has multivariable real analysis, Lebesgue integrals, and differential forms in it. The latter half of Bartle's The Elements of Real Analysis has a lot of the multivariable stuff in it. Royden's book has good coverage of Lebesgue integration, and the best book on differential forms that I know of is the one by Harley Flanders, Differential Forms with Applications to the Physical Sciences. It's aimed at scientists, but it's rigorous enough for mathematicians, too.

 

[SrVishi]

Thanks for your reply. What do you think of Real Mathematical Analysis by Charles Pugh?

 

[WWGD]

I am pretty sure if you go over books under the section of "Advanced Calculus." you should be able to find something. I remember a good one by an author of last name Edwards. Let me see if I can find it.

 

[Geofleur]

I haven't studied Pugh's book, but it looks good from a glance!

 

[mathwonk]

If you like Rudin's chapters 1-8 you might as well read those other chapters too. I have heard many people criticize his treatment of differential forms, but not his Lebesgue integration. Mind you I don't like the way Rudin treats anything, but for you who do like it, his chapter on Lebesgue is probably no worse than chapters 1-8. As I recall it doesn't go very far. I tend to like books by Sterling K. Berberian. I also like the calculus of several variables book by Wendell Fleming, which also treat Lebesgue integration nicely, along with vector calculus.

 

[SrVishi]

Thanks again mathwonk! You say his chapter on Lebesgue integration "doesn't go very far." Do you think this is sufficiently (in terms of coverage) expanded upon in Real and Complex Analysis?

 

[mathwonk]

yes certainly. I don't like that book either but it does contain a lot of material. I just find I never understand anything from Rudin. Congrats to you that you do. My analyst friends like his book so it must have some important virtues. Maybe I just don't understand analysis. Actually i don't understand algebra either, or statistics, or,...

 

[WWGD]

This is the book I was referring to: http://www.cuil.pt/r.php?cx=0028257...-8&q=C.H+Edwards,+advanced+calculus&sa=Search

 

[Michael Tilles]

Spivaks Calculus on Manifolds covers the chapter 9 material well. For Lebesgue theory Follands Analysis text is quite good but he spends quite a bit of time developing general measure theory and then covers Lebesgue measure as a special case. Roydens text does the reverse doing Lebesgue measure on the reals and then redoing all the work for general measures, Rudins Real and Complex is highly non-standard (compared to other texts),in its development of measure theory,however I would still check it out.

 

[SrVishi]

Thanks. Can you let me know how his development is different? It could help when I go through it.

 

[Michael Tilles]

Folland starts with definitions of measures and goes through their construction. The way it goes is you start with some increasing right continuous function which generates a pre-measure which generates an outer measure which you can restrict to measurable sets to get a measure,and much later (Chapter 7?) he proves a form of the Riesz-Representation Theorem which says an positive linear functional on a suitable space of functions is given by integration with respect to a Radon measure. Rudin spends a great deal of time early on to prove the Riesz Representation theorem and uses that to construct measures and derive their various properties. It really depends on where you want to put the up front work in. Do you spend time constructing measures and then use that theory to get information about positive linear functionals or vice versa. Personally, I found Follands approach easier to wrap my head around but it's worth checking both out. I should note that both are very much and abstract approach to measure theory of which Lebesgue measure is just one special case. There are all sorts of measures;Haar measures,radon measures etc.. and Rudin and Folland are both geared at laying out the general theory so that you use and study these. I think Pugh covers everything Rudin does in Principles but with more exposition, and I really enjoyed his multivariable section. I haven't worked through his Lebesgue chapter but based on the table of contents it looks to be more than sufficient. If this is your first exposure to this I don't see any reason to rush into the general measure theory and I think the concrete case of Lebesgue measure is making sense of the theory. If you have access to a good library I would take a look at all three and see what you like.

Also if you are interested in complex analysis,Conway has a great book on the subject. Many people like Apostol but I found it to be rather soulless.

This last bit is purely anectdotal but I don't know anyone who learned measure theory from Rudin's Real and Complex Analysis.

 

[mathwonk]

lebesgue integration concerns identifying a good class of integrable functions that is closed under natural limit processes. there are teo ways to proceed by my understanding (spoiler alert: I am a great novice, totally non expert)

1) develop measure theory first. the theory of sizes of complicated sets, via squeezing them from inside and ou by infinite sequrnces of rectangles, and distinguish those which ahve the same inner and outersize limits.

2) develop a theory of limits of functions of various types, and define lebesgue integrable functions as those that are approximable by certain simpler types of functions, and define the integrals as limits of those integrals.Most books prefer the meaure theory version (1), although it takes more work. I myself dislike almost all books written by Rudiona nd like almost all books written by Sterling K. Berbnerian, both of whom give the measure theory version. My analyst friends like the book by Wheeden and Zygmund, the second named author being a famous analyst. I have not seen it but I would l;ike to get a copy someday. I myself like a book on advanced calc by Wendell Fleming, that treats also lebesgue integrals very clearly.

Another nice treatment is an early chapter of Royden, but most of that book is not that great to me at least.I do not recommend Dieudonne's book foundations of maodern analysis vol 2, which uses the second version 2) in a very unintuitive construction of lebesgue integrals.Words to the wise: since I do not myself understand integration theory but my analyst friends do, go with the book by Wheeden and Zygmund, which they recommend.

 

[mathwonk]

read at your own risk:

The first concept is essentially due to Riemann, the idea of a set of measure zero.

a subset of R has measure zero if, for every e>0, it can be covered by a sequence of intervals whose lengths have infinite sum < e.

then two functions are considered equivalent for the purpose of integration theory if they agree except on a set of measure zero.

In particular two equivalent functions should have the same integral over any interval.

E.g. all countable sets have measure zero, hence the constant function 1, is equivalent to the characteristic function of the irrational numbers, i.e. the function equal to 1 on irrationals and equal to 0 on rationals.

The outer measure of a bounded set is the inf of the numbers obtained by summing lengths of sequences of intervals which cover the set.

Then a bounded set is measurable if in every interval, its outer measure plus the outer measure of its complement equals the length of the interval.

a function is measurable if the inverse image of every measurable set is measurable.

a sequence of functions converges to another function “almost everywhere” if it does so at all points off a set of measure zero.\

the integral of a measurable function is define analogously to the Riemann integral except using measurable sets in place of intervals. i.e. subdivide the interval of values into subintervals of length e>0, and define the upper and lower integrals for that subdivision as the right and left hand values of those subintervals multiplied by the measure of the inverse image of those intervals. then the integral is the common limit of those upper and lower integrals if they do agree.

Then if a sequence of measurable functions converges almost everywhere to a measurable function, the integrals should also converge.