Talk:Standard probability space

I'm not familiar with the idea of "countable probability spaces" being "standard". Is there any reference to this idea in a text book?

The definition is given according to the three articles cited (and many others). About textbooks, I'll try to find something appropriate. Boris Tsirelson (talk) 08:51, 30 January 2008 (UTC)[reply]

Not a text book, but an encyclopedia is found. Namely: Encyclopedic Dictionary of Mathematics (second edition) by the Mathematical Society of Japan, edited by Kiyosi Ito, The MIT Press. See Volume 1, article 136 "Ergodic Theory", item A "General remarks" on page 531:

A Lebesgue measure space with a finite measure (σ-finite measure) is a measure space that is measure-theoretically isomorphic to a bounded interval (to the real line) with the usual Lebesgue measure, possibly together with an at most countable number of atoms.

Boris Tsirelson (talk) 10:16, 30 January 2008 (UTC)[reply]

An appropriate textbook is found, and added to the article as [4]. In the book, see Item (a) on page 65 (just after the proof of Theorem 2.4.1). Boris Tsirelson (talk) 19:30, 30 January 2008 (UTC)[reply]

Article needs major rewrite

The first two "pro & con" sections should either disappear or be moved under the section called "Using the standardness". This is not an article about legalizing abortion or drugs; it's just a mathematical concept.
The attempt to define "pathologic" might go a lot easier if a standard probability space were defined first and then a probability space were said to be "pathological" if it is not standard. A section giving some motivation for a formal definition of standard probability space would be appropriate in its place, but it is futile to attempt to exhaustively cover every case that might be considered pathological or "non-standard" in some vague undefined sense.
The formal definition(s) of a standard probability space should be given more prominence in the article, and perhaps elaborated upon some more. This is, or should be, after all, the main point of the article.
History should go after the definition. I want to know what it is before I care to read about its history.
The general tone/style of writing may be a little too informal in a few places. (A very minor issue here.)
Contributors should try to avoid citing their own works (unless, of course, they are published in a peer-reviewed journal.)

Deepmath (talk) 02:05, 26 August 2008 (UTC)[reply]

Thank you for the criticism; I'll think on it. Boris Tsirelson (talk) 07:02, 26 August 2008 (UTC)[reply]

I did, according to the opinion of Deepmath, in spite of my suspicion that some wikipedians think differently. Well, they may participate, too. Boris Tsirelson (talk) 12:32, 26 August 2008 (UTC)[reply]

By all means, please do voice your reservations and/or objections to any of my criticisms. I apologize if I sounded too bossy at first. As the article was before, I read through an introduction, pros and cons, some history, and a rather deep discussion of certain pathological counterexamples in probability theory before I even got to the definition. Some gentle introductory material before the technical content is certainly advisable in an article intended to be useful for the general public, i.e. non-mathematicians. A succinct but precise definition for the general reader should probably go right at the beginning of the article, like so:

In probability theory, a standard probability space is a probability space satisfying certain assumptions of "standardness" introduced by Vladimir Rokhlin in 1940.^{[citation needed]} He showed that the Lebesgue measure on the unit interval is sufficient as a probability space for all practical purposes in probability theory. The existence of the Wiener process implies that the dimension of the unit interval is not a concern: Lebesgue measure in any countable number of dimensions, in the product topology, can be successfully mapped, or pulled back, to a single dimension....

If you start off more or less like this, even a completely non-mathematical reader will know and be able to say what a "standard probability space" is just from reading the first sentence. Then you can explain the history in more detail immediately after the introduction (more like you had it before) if you want. The formal definitions can go later, then. And on second thought, I should probably take back what I said about trying to define what is meant by "pathological"—trying to identify and exclude these cases was no doubt precisely the reason for defining the notion of "standard probability space" in the first place. I certainly don't want my suggestions to make the article worse. The general outline could go like this:

Intro (including a succinct informal definition)
History (including some of the pro/con information in order to explain why standard probability spaces have received limited study in the West)
Motivation (including some discussion of the pathological cases we want to avoid)
Formal definitions (more than one would be nice)
"Using the standardness" (including more detailed pro/con information: what benefits and advantages are realized and at what cost, more about the pathological cases we are avoiding, etc.)
References, Further reading, See also, External links, Categories, etc.

(I know that's more like how it was before than how it is now after following my suggestions.) This is a large topic, and it could use a rather comprehensive article. Deepmath (talk) 00:13, 27 August 2008 (UTC)[reply]

Now I have nothing to voice. My doubts (reservations, objections) are already voiced by you! Your new suggestions do not raise new doubts. I try to implement them. Boris Tsirelson (talk) 17:30, 27 August 2008 (UTC)[reply]

The next iteration is made; please look. I know that something remains to do (especially, some more equivalent definitions). At this point, however, let me remind to you and all, that I am just the initiator, not the owner of the article. Start making your improvements yourself! Boris Tsirelson (talk) 19:40, 27 August 2008 (UTC)[reply]

I'm going to take a break for a while, but I should be back next week. I'm still trying to get a better handle on completeness vs. regularity for probability measures, which seem to be mutually exclusive on a separable space, unless I'm not understanding something quite correctly. I'll have to do some more research so I actually know what I'm talking about before I can actually contribute much to this article. For example, I'm suspecting that if a measurable space admits a complete measure, it might in fact have "too many" measurable sets in its σ-algebra to have the regular conditional probability property. But there might be some sort of correspondence I'm overlooking here, and there almost certainly is, if you insist that a standard probability measure be the completion of a Borel measure. I did change the remark you made to that effect in that article, but I think (I know) it's still not right, so feel free to fix it. Deepmath (talk) 00:35, 28 August 2008 (UTC)[reply]

Yes, I fixed it. You should not just say that a measurable space is Radon, for a trivial reason: A measurable space itself is not a topological space, while to be Radon or not is defined for topological spaces only. Boris Tsirelson (talk) 08:01, 28 August 2008 (UTC)[reply]

Completeness vs. regularity for probability measures

"...seem to be mutually exclusive on a separable space..." --- Why? If regularity holds before completion, then it still holds after completion. Indeed, you add/subtract a null set; and the null set is included into an open set of arbitrarily small measure... Boris Tsirelson (talk) 08:14, 28 August 2008 (UTC)[reply]

"...I'm suspecting that if a measurable space admits a complete measure, it might in fact have "too many" measurable sets in its σ-algebra to have the regular conditional probability property..." --- Yes, you are right. The Lebesgue σ-algebra (on R, or [0,1]) indirectly reveals the σ-ideal of null sets. (Namely, a set is null if and only if all its subsets belong to the Lebesgue σ-algebra. Of course, I mean, in ZFC; see the last paragraph in non-measurable set.) Thus, every nonatomic measure on the Lebesgue σ-algebra is absolutely continuous (w.r.t. the Lebesgue measure). However, conditional measures typically are singular (with no atoms and no absolutely continuous part). This is why one defines them on the Borel (not Lebesgue) σ-algebra.

More generally: analysts often work with the Lebesgue measure only (and absolutely continuous measures) and therefore prefer the Lebesgue σ-algebra (to the Borel one). In contrast, probabilists often work with many mutually singular measures and therefore prefer the Borel σ-algebra (to the Lebesgue one). Boris Tsirelson (talk) 08:34, 28 August 2008 (UTC)[reply]