Is Ergoff's Theorem Vital to Understanding Measure & Integration Theory?

[Nusc]

What's a well written textbook on this topic? A textbook that has about the same exposition as Munkres Topology.

 

[leon1127]

halmos

 

[eastside00_99]

oh, god! I am taking a class and we are using Friedman. I have never disliked a math course so much. Here is a theorem:

Let X be a finite measure space. If a sequence {f_n} of almost everywhere real-valued, measurable functions converges almost everywhere to an almost everywhere real valued measurable function f, then {f_n} converges to f almost uniformly.

The pendantics astound!

 

[zhentil]

What on Earth are pendantics? That theorem is non-trivial and very useful, if you ask me.

I'd recommend Wheeden and Zygmund. The exposition is terse, but they build up from R^n using a geometric approach, which makes life much easier than the standard "outer measure is a set function satisfying the following" approach.

 

[George Jones]

The course I took as a student used Bartle's The Elements of Integration, and I quite like it. It has been bundled together with The Elements of Lebesgue Measure and reissued as http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471042226,descCd-tableOfContents.html" .

 

[mathwonk]

In my youth i never found a book i could really learn this topic well from. i tried halmos and found it extremely dry and formal. i tried lang's analysis and kind of liked it but still had a very hard time. i tried m.e.munroe but again found it pretty abstract.

as little as i care for rudin's undergrad book, the grad version, real and complex analysis may be better.

my favorite, at least in places, was riesz and nagy.

years ago many people also liked the first chapter of royden, but again i read it and still felt little connection to the material.

more recently some of the experts in my department, who really understand the subject, seem to like wheeden and zygmund.

 

[eastside00_99]

What on Earth are pendantics? That theorem is non-trivial and very useful, if you ask me.

I'd recommend Wheeden and Zygmund. The exposition is terse, but they build up from R^n using a geometric approach, which makes life much easier than the standard "outer measure is a set function satisfying the following" approach.

pedantic: it is a adjective, and it means to be overly concerned with minute details.

I used the word as a noun which may or may not be "allowed."

I didn't say anything about it being trivial or a waste of paper!

 

[las3rjock]

pedantic: it is a adjective, and it means to be overly concerned with minute details.

I used the word as a noun which may or may not be "allowed."

I didn't say anything about it being trivial or a waste of paper!

Welcome to real mathematics! Minute details matter!

 

[morphism]

I have to agree with mathwonk. There doesn't seem to be any really good measure theory books out there. I've tried sampling a lot of them, and never found one that made me think "Wow, this is really well-written!".

That said, I second George's recommendation of Bartle. Royden is also alright, but I found it annoying that he leaves lots of important results to the exercises (e.g. Egorov's theorem, Lebesgue's criterion for Riemann integrability, etc.). It isn't a great text, but if you read it and do the exercises, you will learn measure theory. It also covers a lot more than Bartle.

 

[eastside00_99]

Welcome to real mathematics! Minute details matter!

What is fake mathematics? No, I know they matter. I mean this is Ergoff's theorem or something which means it must be important. Its just I prefer geometry to analysis, and have realized that the only reason I liked functional analysis last semester is because I liked considering things like functions as points. Maybe once I take multivariable complex analysis, this stuff will come alive to me. Anyway, I think I am allowed to not like a theorem or a subject no matter how important it is and still be a member of this real mathematics club.