Is integration theory supposed to be this hard?

[Fredrik]

Is that the same as Cauchy in the L1 norm?

Yes, I think Friedman's avoiding that term because this is earlier in the book than the definition of the L^p spaces.

Edit: I thought about it some more, and I have to change my answer to "no". These concepts are almost the same, but the L¹ norm only applies to bounded real- or complex-valued functions, while these are extended real-valued functions (that are almost everywhere real-valued).

 

[Fredrik]

Lang's Analysis II (maybe now Real analysis), has a good strong statement of Fubini. (And the functions in Lang have values in any Banach space.)

Thanks for this tip. I just checked it out. It looks really good. The actual title is "Real and functional analysis". Lang is using the same definition as Friedman, but starts with complex-valued functions right away (and doesn't use any properties of ℂ other than the ones shared by all Banach algebras) This is how he explains his choice to use the limit definition in the introduction to the chapter:

A posteriori, one notices that the monotone convergence theorem and the "Fautou lemma" of other treatments become immediate corollaries of the basic approximation lemmas derived from Lemma 3.1. Thus it turns out that it is easier to work immediately with complex valued functions than to go through the sequence of many other treatments, via positive functions, real functions, and only then complex functions decomposed into real and imaginary parts. The proofs become shorter, more direct, and to me much more natural. One also observes that with this approach nothing but linearity and completeness in the space of values is used. Thus one obtains at once integration with Banach valued functions.​

I'm going to have to read more of it. It looks like a version of what Friedman did, that's just better organized and with proofs that are easier to follow.

 

[micromass]

Thanks for this tip. I just checked it out. It looks really good. The actual title is "Real and functional analysis". Lang is using the same definition as Friedman, but starts with complex-valued functions right away (and doesn't use any properties of ℂ that other than the ones shared by all Banach algebras) This is how he explains his choice to use the limit definition in the introduction to the chapter:

A posteriori, one notices that the monotone convergence theorem and the "Fautou lemma" of other treatments become immediate corollaries of the basic approximation lemmas derived from Lemma 3.1. Thus it turns out that it is easier to work immediately with complex valued functions than to go through the sequence of many other treatments, via positive functions, real functions, and only then complex functions decomposed into real and imaginary parts. The proofs become shorter, more direct, and to me much more natural. One also observes that with this approach nothing but linearity and completeness in the space of values is used. Thus one obtains at once integration with Banach valued functions.​

I'm going to have to read more of it. It looks like a version of what Friedman did, that's just better organized and with proofs that are easier to follow.

Aah, that is very interesting. This also shows that Friedman's treatment of the integral is superior than my idea of "take the supremum". Indeed, Friedman's definition can be generalized to Banach spaces, while my definition cannot. Indeed, there is no notion of supremum in a general Banach space. (you'll need a Banach lattice for that).

 

[pwsnafu]

Thanks for this tip. I just checked it out. It looks really good. The actual title is "Real and functional analysis". Lang is using the same definition as Friedman, but starts with complex-valued functions right away (and doesn't use any properties of ℂ other than the ones shared by all Banach algebras)

This is how I was taught.

 

[kai_sikorski]

Haha, I'm visiting MBI for the semester. Just realized he's in the office across the hall from me.

If the curiosity is so high, you can even try to ask him :)

http://www.math.osu.edu/~friedman.158/

Email: afriedman@mbi.osu.edu (shown in public)

Varför inte?

Note: he has so incredible CV with plenty publications!

 

[Fredrik]

It's amazing that he got his Ph.D. 56 years ago and is still active. No need to ask him any questions on my behalf though. I got most of it figured out by now. The book by Lang explains some of the things I was wondering about, and helped me figure out a few more. I will however retract my comment that Lang's presentation is better. Some things are clearer in Lang. Some things are clearer in Friedman. I'm glad I have access to both.