Talk:Baire function

Page based on a misunderstanding?[edit]

This page seems to have been provided in response to a request on Wikipedia:Requested articles/mathematics; someone asked at the same time for articles on Baire set and Baire function. Unfortunately these are ambiguous terms, but I suspect that what he was looking for was a definition where a Baire subset of a topological space X is an element of a sigma-algebra that I've seen in some sources as the smallest sigma-algebra containing all compact G_δ sets, and in others as the smallest sigma-algebra containing all continuous preimages of half-open intervals in the real line. If X has sufficient niceness properties (I think T₄ plus sigma-compact is enough) then the Baire sets are identical to the Borel sets, but in general they may not be.

In this context, "Baire function" has a definition that I can't seem to locate at the moment, but it's one that implies that Baire sets are precisely the ones with Baire characteristic functions. I don't see any reason to think these are the same as Baire class α functions for any particular α. It would be nice if someone could straighten this out. Note that Baire set will have to be a disambiguation page, as the term can also mean "set with the property of Baire". --Trovatore 05:56, 6 February 2006 (UTC)[reply]

Ah, here it is:

A family of functions defined on a space R is called complete if the limit of any sequence of functions in it is also contained in it. The minimal complete family of function defined on R which contains all continuous functions is called the family of Baire functions and its element a Baire function. A subset of R is called a Baire set if its characteristic function is a Baire function.

Nagami, Keiô (1954). "Baire sets, Borel sets and some typical semi-continuous functions". Nagoya Math J. 7: 85–93.

So in reading the above it's important to notice that Nagami is talking about continuous functions from a space R to the reals; R does not mean the reals. That's an odd choice, but so be it. In any case it seems that a "Baire function" in this sense is one that is Baire class α for some α. In that case the article is not in as bad shape as I thought. However it would be nice to check whether this definition of Baire set matches the one about the smallest sigma-algebra containing the compact G_δ's–the ref being from 1954, there could well have been refinements since then. --Trovatore 06:28, 6 February 2006 (UTC)[reply]

I am fairly sure that this page is not actually based on a mis-understanding, but it does belong to a slightly out-of-fashion branch of measure theory. I can vaguely recollect seeing definitions of such a hierarchy of Baire functions many years ago, but unfortunately cannot recall the exact details. Madmath789 15:57, 29 May 2006 (UTC)[reply]

The hierarchy I know about. The question was whether the union of this hierarchy matches the collection of functions that have Baire preimages of open sets, in this peculiar sense of "Baire" meaning "element of the smallest σ-algebra containing the compact G_δ's". --Trovatore 16:08, 29 May 2006 (UTC)[reply]

I am now convinced (from Nagami's paper, Doob's Measure Theory, Halmos's Measure Theory and Springer's online encyclopedia) that the two senses of "Baire" do really coincide. However, I believe that the article as it stands is not fully correct, in that the Baire function hierarchy has to be defined not just for natural numbers, but for infinite ordinals.Madmath789 16:38, 29 May 2006 (UTC)[reply]

I think i know where this confusion comes from. This page correctly describes the 'Baire classes' as the hierarchy of function classes, where Baire class 0 is the continuous functions, and each subsequent class is the set of functions which are pointwise limits of the functions in the previous class. To describe these as the Baire functions makes some kind of sense. The use of the term 'Baire set' for those sets whose indicator function lies in some Baire class, I am unfamiliar with, also the characterisation of this set as a sigma-algebra. However, a different notion is that of a 'property of Baire' set. Such a set can be expressed as the difference of an open set and a first category set. These sets do form a sigma-algebra. The property of Baire functions are then those for which the preimage of any open set is a property of Baire set. The property of Baire functions and the functions in Baire class 1 can be characterised in a way which superficially looks similar, but is in fact different. This information I am taking from the book 'Measure and Category' by Oxtoby. I suggest that adding the two articles 'Property of Baire function' and 'Property of Baire set' will prevent people confusing those notions with this. Also, can we check that the use of the terms 'Baire function' and 'Baire set' does in fact refer to what's talked about in this article? Via strass 01:08, 31 May 2006 (UTC)[reply]

The property of Baire article already exists; "Baire set" is a legitimate term for "set with the property of Baire", and also for this different notion. That's why Baire set is a disambig page (albeit with a redlink). --Trovatore 05:17, 31 May 2006 (UTC)[reply]

Off-topic note: I'm not happy with the profusion of the term "indicator function" in the math project. Yes, it's unambiguous and that's good, but only the probabilists know what it means, and that's bad. I think we should stick to "characteristic function", on the grounds that it's the overwhelmingly more used term, and disambiguate in situ when confusion could arise. --Trovatore 05:22, 31 May 2006 (UTC)[reply]

I would agree wholeheartedly with this - it would make more sense for an encyclopedia to use the terms understood by the majority of its readers. Madmath789 07:05, 31 May 2006 (UTC)[reply]

Class 3[edit]

Can someone add an example of a Class 3 function? Eric Kvaalen (talk) 18:58, 1 June 2013 (UTC)[reply]

I'm afraid, this is rather hard; see [1], and maybe the first item in [2], and Theorem 2.7 in [3], and the last page in: H.H.Pu, H.W.Pu and T.H.Teng, "Symmetric and approximate symmetric derivatives for symmetric functions" (1979), and [4].Boris Tsirelson (talk) 11:30, 20 August 2014 (UTC)[reply]

Really, there is an old good way to construct a class n set for every finite n (and even not finite...); transition from sets to functions should not be hard (relatively...). Some explanation of that way are available in Sect.1c of this course. However, for class 1 and class 2 we have much simpler examples. The question is, whether for class 3 we still have a "special" example much simpler than the "general" one. Boris Tsirelson (talk) 16:43, 20 August 2014 (UTC)[reply]

See "On a hierarchy of Borel additive subgroups of reals" by Ashutosh Kumar, a preprint of 2009 available here. For now, not a reliable source... Boris Tsirelson (talk) 14:14, 21 August 2015 (UTC)[reply]

See also Sect. 3 in Kumar's thesis available from the site of Steffen Lempp. Boris Tsirelson (talk) 14:44, 21 August 2015 (UTC)[reply]

Class 2[edit]

The Cantor set example is wrong (it is of Baire Class 1, not 2); I tried to give the right example, but somebody, without any discussion, reverted it back to the wrong example. The right example is in Oxtoby's book, p. 33. Manta (talk) 13:26, 23 January 2014 (UTC)[reply]

Also here. Boris Tsirelson (talk) 11:17, 20 August 2014 (UTC)[reply]