Difference between revisions of "User talk:Boris"

Revision as of 18:04, 8 April 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 "resells" 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)

First I must say thank you, I really appreciate the discussion. Especially on something so bread-and-butter (common/trivial) for you. Please don't see this as arrogance when I say I'm still leaning towards "ring", it is very much a programmer thing I thing to be fighting two opposites, usually "performance" and "abstract" or the "concrete" and the "abstract". As far as I can see I can distill the basics of measures into something true on (sigma)rings. As every algebra is a ring, I've abstracted part of the algebra definition, but I can abstract no further and have the "measure" properties I want.

This occurs a lot, "category theory" distills things to "arrows between objects" for example, using [ilmath]A\prec B[/ilmath] to mean "[ilmath]A[/ilmath] is less abstract than [ilmath]B[/ilmath]":

1. Inner product [ilmath]\prec[/ilmath] Norm [ilmath]\prec[/ilmath] Metric [ilmath]\prec[/ilmath] Topological space
2. Abelian group [ilmath]\prec[/ilmath] Group
3. The whole "chain" from ring to field

So forth. This is a very powerful argument. I do however see your point, that large (huge!) chunks of measure theory (that I know of) deal with [ilmath]\sigma[/ilmath]-algebras, for example:

• Probability, we require that [ilmath]\mu(X)=1[/ilmath], to do this [ilmath]X[/ilmath] must be in the ring of sets, but if [ilmath]X\in[/ilmath] the ring then it's an algebra of sets, thus Probability only deals with algebra

Every space I imagine also is an algebra.

Rings do sidestep one problem though, which I also like, if I start talking about the "sigma algebra generated by" some small intervals like [ilmath][0,\tfrac{1}{4})[/ilmath] and such, it becomes very blurred as to what "the entire space" is, is it all of [ilmath]\mathbb{R} [/ilmath]? [ilmath][0,1][/ilmath]? So forth. This is not a very convincing argument though.

I've found a PDF of the book you mentioned and I shall certainly check that out!

I really like the "buyer/seller" analogy, Abelian groups are extremely close to just normal groups, and normed spaces are extremely far from metric spaces (they've got an entire vector space structure on them!) an algebra is not too close to a ring, but not that far either. When creating the pages, if I do choose rings, I shall be very clear that one must keep "sigma-algebras are rings" at the forefront of his mind.

Lastly, I'm sorry for the long reply. I hope we (I) am not making a mountain out of a molehill and I truly value greatly your input. Alec (talk) 00:30, 25 March 2016 (UTC)

A boolean algebra (I know these, they're in a trivial (dare I say canonical) bijection with sigma algebras) of boolean algebras. Wow. This is why I love abstraction.

Well, in the natural tension "developers-users" you take the developers' part. That is your right. Some readers will like it, some will dislike it. In math it is often called "Bourbaki-ish". In programming, rather often, developer makes a lot of options, which annoys a user. An example: ImageMagic. I am thankful to Dave Newton for the "smartresize" script. He spent a lot of effort for choosing a lot of parameters of ImageMagic in a very useful and typical combination, and he saved me (and many others) a lot of time. Boris (talk) 07:26, 25 March 2016 (UTC)

Sigma-ideal source & "formal" probability book recommendation

Hello again, I'm glad to see you're still around and I hope you're well, and sorry about the recent "lull" in activity, it's due to the thing I mentioned in the email and the bad patch is passing now! I've searched for Sigma-ideal and from what I can tell:

• [ilmath]\sigma[/ilmath]-ideal - the name given to a hereditary system of sets that is also a [ilmath]\sigma[/ilmath]-algebra

However I cannot find a source! I don't doubt you but I'd like something in writing (even if you wrote it!). If I recall correctly, you recommended a book once before "generalised measure theory" and that doesn't have it in. I'm also seeking a probability book. I have a few but they don't deal with the formalities of measure theory and my measure theory books just mention "fun fact, a probability space is.... a random variable is...." and so forth. There was a book called "probability and measure" by a famous author (it was its 10th edition) that also lacked the probability and measure I was hoping for. Although I suppose there isn't much to say. Alec (talk) 17:17, 8 April 2016 (UTC)

Hmmm, surely not "a hereditary system of sets that is also a sigma-algebra"; being a sigma-algebra it contains the whole space, and then, being hereditary, it contains everything (which is too boring). I wondered where did you find it; on Wikipedia I found something a bit unexpected to me: Sigma-ideal; but there, the sigma-ideal is contained in sigma-algebra, which does not mean that it is a sigma-algebra. They define (according to a book by Bauer) a sigma-ideal in a given sigma-algebra; and it is not quite hereditary. What I mean is their sigma-ideal in the sigma-algebra of all subsets. Really, they are a bit inconsistent; their sigma-ideal is generally not their ideal.

Here is my source (a very authoritative book on descriptive set theory):

Alexander S. Kechris, "Classical descriptive set theory", Springer.

There, page 41 (Sect. 8.A): "An ideal is..." (equivalently to WP); "If it is also closed under countable unions it is called a sigma-ideal." Boris (talk) 17:41, 8 April 2016 (UTC)

I never thought to look in a book with "set theory" in the title. I have this massive book by Thomas Jech just called "Set theory", they're also defined in there. Unfortunately my library do not offer a digital copy of that book (they do with most Springer stuff.... weird) nor can I find a PDF; but I have found the relevant parts on Google Play. I will have a Set Theory exam and I think it'll be good to get something that wasn't written by Jech in. If you have the book though it must be a good reference. Thank you for this. Alec (talk) 18:04, 8 April 2016 (UTC)