BTW, set theory literature update: I've directed Amazon to send me a copy of Kunen's book. They had Cohen's Set Theory and the Continuum Hypothesis for £12, so I threw that in also.
In fact I think I have a photocopy of Cohen lying around somewhere, from when I was an undergraduate and took a masters-level formal logic course. I must have asked irritating enough questions that the prof donated a copy to me ...
... however, I was young and uneducated and couldn't abide the notation, using \supset for implication. So I gave up on it when I found there was a page spread missing from the photocopy.
Anyone know of a reason behind the use of \supset for logical implication, or was it just a coincidence that the same symbol came to be used for two more-or-less opposing concepts?
@HenningMakholm Who uses the supset notation for implication? (This is a serious question btw.) Also is there a different LaTeX command for that symbol in this context, or do you actually say "A \supset B" to mean logical implication? That looks a little awkward.
@tb There aren't that many people posting questions so often.
@AsafKaragila Sounds more existentialist than nihilist to me, but what do I know.
Interestingly enough, Peano's table implies that he was using e.g. $x \epsilon N$ to mean $x\in\mathbb{N}$, but glossing \epsilon as "is" and N as "a positive integer".
Does someone know if Grafakos in his harmonic analysis books assumes that the measures are sigma-finite? I can't seem to find it but he talks about simple functions, and I don't think they are dense if the space is not sigma-finite.
I think he tries to correct the statement of Riesz-Thorin which I wrote down to add "sigma-finite" and then an arrow to "simple function" where he writes "not dense".
You only need (something like) sigma-finiteness in order for the duality between L^1 and L^\infty to work well. I'm pretty sure interpolation techniques should work in greater generality.
I wrote down some of my results not in full generality because I didn't need them in full generality, but the proof needs only small modifications to have a more general situation (say from p = 2 to 1 < p <= infty). My advisor then corrects it. What is best? Full generality or not?
I had also cited a theorem inside a proof (which I only need there), but there were some complaints about that. Then I said that Stein does it as well. Then he was more like: Well, if Stein does it, I can't say it is wrong, but I like it better if you cite it before the proof. Oh well.
Sure, just cite it before the proof. I liked it better this way because it creates a continuous story, but if the one that has to give me a grade dislikes it...
@Jonas: another solution would be to paraphrase: We want to prove that ... . This follows from Stein's Theorem X, provided we check that a) ... b)... c)... hold.
"I know at least one poster here that'll jump at the chance if any of it is erroneous......" - I wonder if that poster is me. I have asked em for supporting references in the past...
Well, that would make 2 possibly 3 sentences that would easily fit in a comment (and they aren't more because they are not backed with references) and KCd asked for earlier attributions than 1965:
"I found the name "Fubini's theorem" used for multiple Riemann integrals in Spivak's Calculus on Manifolds (1965). Does anyone know of an earlier usage of the label "Fubini's theorem" for multiple Riemann integrals?"
He doesn't mention any of those. Anyway, I'm not voting, someone else already did...
KCd's post asks for facts (the earliest references that associate the theorem with Fubini), not why the switch happened (this, we might never know for certain).
@Srivatsan: No, completely wrong. Metric spaces: 1906 Fréchet; Hausdorff spaces: 1914 Hausdorff; general topological spaces: 1922 Kuratowski, more history here
@Srivatsan: of course, there were attempts to capture the notion of convergence in so-called convergence structures earlier on, but that's also around the turn of the centuries.
@Srivatsan Well, he modeled his things à la Riemann by coordinate patches, roughly speaking. The precise definitions were quite fuzzy. Only Weyl clarified the notion of surfaces, and the notions of manifolds, polyhedral complexes etc. came only in the 20ies. Rado, Alexandroff, Reidemeister, etc.
Nowaday's definition of a manifold goes back to Whitney in the 30ies, if I'm not completely mistaken.
Also, Poincaré first stated it in terms of Betti numbers which he then observed to be wrong with his "fake sphere". Only then he introduced the notion of the fundamental group.
@Srivatsan I believe it's a matter of the culture you grew up in. The human brain is incredibly good at adapting to training and if you're not biased towards nowaday's standards of precision you don't even observe that things are fuzzy.
@tb: I have a script that converts × to ×, but otherwise I just type it. If there are too many exponents or if it is a displayed equation, then I use mathjax. Roughly I just follow the manual of style from wikipedia. Its very easy once you get used to it. It doesn't take any longer than dollar signs, and never hangs the browser while editing, etc.
