Difference between revisions of "User talk:Boris"

Revision as of 19:01, 24 March 2016

You seem to find all the oldest pages!

This project started on the 12th of Feb 2015 and it seems you've found all the oldest pages! Back when I didn't really know how to use a wiki! If you do find any that are bad, please mark them with {{Dire page}} this makes them a priority for being turned into "stub pages", stubs marked with {{Stub page}} - this means they're short and need fleshing out but provide some useful information. Lastly, if you encounter anything with 1 reference, or few references (for a large bit of content) please mark it with {{Requires references}}. The other to-do markers are:

• {{Todo}} - for small things, like add a few links, minor change
• {{Requires proof}} - some things are missing proofs.

In the old pages EVERYTHING uses the {{Todo}} template, which has given the To-do category a lot of clutter.

Lastly (on the note of marking), all these to-do templates support a comment, so for example you can use:

• {{Requires proof|Simple, just show {{M|X\implies Y}}}} say.

Measure theory is the oldest area (that's what I was working on at the time), as such I've started Site projects:Patrolling measure theory which is a snapshot of the entire measure theory category as it stood when the project started. This should ensure a minimum standard of quality across the site. Alec (talk) 20:56, 19 March 2016 (UTC) PS: I hope you've seen some good pages!

I just took the first pages listed on Category:Definitions. Do you want me to do differently? Which way? I do not have a good orientation on this site, of course. Boris (talk) 21:05, 19 March 2016 (UTC)

Ah of course, alphabetically! I was joking about how you've found some of the oldest pages here! There's no wrong way to browse. Alec (talk) 21:23, 19 March 2016 (UTC)

Some questions.

It just occurred to me that I can ask questions, you don't have to answer of course. There are a few things I haven't been able to work out.

1. When a measure is "continuous from below", continuous with respect to what?
2. Regarding Addition of vector spaces, I am not experienced enough to decide on a notation to stick to (and there are some contradictions, as noted on the note page), what do you take:
   1. [ilmath]\bigoplus[/ilmath]
   2. [ilmath]\bigoplus^\text{ext} [/ilmath] / [ilmath]\boxplus[/ilmath]
   3. [ilmath]\sum[/ilmath]
   4. [ilmath]\prod[/ilmath]

   to mean (Especially over arbitrary families)? I want to commit and write the page, but I'm afraid to do so until I understand the definitions and how they differ (even if there are trivial/canonical isomorphisms between them). Anything without a warning or not in a notes page is supposed to be trust-worthy.

I am surprised and quite pleased that there are only 2! Alec (talk) 23:28, 19 March 2016 (UTC)

1. The measure is continuous w.r.t. the monotone convergence of (measurable) sets. It means, the limit of an increasing sequence of sets is (by definition) their union. A more general notion: [ilmath]A_n\to A[/ilmath] when the sequence of their indicator functions (in other words, characteristic functions) converges pointwise to the indicator of A. If you want to see a topology (not just convergence), well, it is the product topology on the product of two-point sets {0,1} over all points of the given space (transferred from indicator functions to sets). But, alas, sequential continuity is far not the (usual) continuity (since the product space is not metrizable).
2. No, sorry; I am a probabilist, far not algebraist; I never teach algebra, I do not keep algebra textbooks on my shelf, and I have no opinion. I only could go to the library and browse, but you in Warwick can do it equally well. In the theory of Hilbert spaces I see "[ilmath]\bigoplus[/ilmath]" and never the others, but this is not an argument. Boris (talk) 12:16, 20 March 2016 (UTC)

Measure Theory terminology

I was reading again a few nights ago and suddenly it became rather obvious what was going on. I've settled on this Notes:Measure theory plan terminology and I wonder what you think before I fully commit Alec (talk) 22:20, 23 March 2016 (UTC)

Well, tastes differ, and if you prefer the approach of the Halmos book (or Bogachev, or whoever), this is your right.

As for me, I do not need the notions of "ring" and "sigma-ring". I am completely satisfied with "algebra of sets" and "sigma-algebra". It seems to me that this is the current trend: rings of sets go out of fashion. The (rather authoritative) book "Foundations of modern probability" by Ovav Kallenberg (Springer, 2002) does not use rings.

A word of philosophy. Measure theory (as every theory, and not only theory) has its "developers" and "users". Developers, naturally, want to "sell" more; users want to "buy" less. I am a user, not developer, of measure theory. I'd say, its power users are, first, probability theory and geometric measure theory, and second, functional analysis and descriptive set theory.

My undergraduate course of Lebesgue integration "buys" even less than Kallenberg: outer measure is introduced on page 15, used sparingly on pages 16-17, and never mentioned afterwards. And I was glad to hear from a colleague that students were surprised by apprehensibility of my presentation. I guess, just because I do not press them to "buy" more new notions than necessary.

Why at all rings were in fashion? I do not know; I guess, in order not to use the infinity as a possible value of a measure. For a probabilist, this is irrelevant: probability measure is finite anyway. Sometimes even a probabilist uses a sigma-finite measure; but still, what is a problem? Just say "measurable set of finite measure" when needed. Geometric measure theory uses non-sigma-finite measures (such as Hausdorff measure); and nevertheless, WP article does not use rings (but uses outer measures).

Probability theory uses sigma-algebras differently from analysis. In analysis, the only point is, to distinguish between "tractable" sets and "pathological" sets. Accordingly, a single sigma-algebra is usually enough in analysis. In probability, sigma-algebra also expresses partial knowledge; accordingly, there is conditional expectation on a sigma-algebra. In a game with 100 players, each has its own partial knowledge; thus, 100 sub-sigma-algebras appear. And in fact, one recent article of myself is entitled "Noise as a Boolean algebra of sigma-fields". (By the way, probabilists often prefer "sigma-field" to "sigma-algebra".) But I could not imagine anything like "Boolean algebra of sigma-rings".

Boris (talk) 19:01, 24 March 2016 (UTC)