Talk:Baire set

Improvements to the article
I have never heard of this concept before reading the article, let alone know something about it. Although I am somewhat certain that this concept is "low priority" in analysis, it seems that there maybe some importance associated to it in the context of mathematics. Does anyone know of any published results regarding this concept, or know of its importance in the context of mathematics? Any expert on this concept would be a potential Wikipedian who could help improve this article. It would be great if such an expert were to see this comment an help in developing the article. -- PS T  06:50, 24 April 2009 (UTC)


 * I can add that it is useful when dealing with probability measures on the product of uncountably many copies of (say) [0,1] (which is evidently relevant to random processes). The consistent system of finite-dimensional distributions leads naturally to a probability measure on this sigma-algebra. Whether it can be extended to the sigma-algebra generated by all open sets is a more delicate question. Boris Tsirelson (talk) 11:01, 24 April 2009 (UTC)
 * Thankyou very much for your comment. It would be great if you were to add this to the article, with references, as I am in no (able) position to do so. -- PS T  13:43, 24 April 2009 (UTC)
 * Thankyou very much Boris! The article looks great now, and I belive it to be no longer a stub. Once again, thankyou - this article would still be a stub were it not for your edits.
 * I am glad that you like it. The style could be polished more, though. Boris Tsirelson (talk) 06:26, 26 April 2009 (UTC)

"Explained example of a Borel set not being a Baire set"
To User:Tal physdancer: No, sorry. An open set, in general, is not completely determined by countably many factors. The reason is simple: a single-point set is closed; its complement is open; and neither is completely determined by countably many factors. (All the factors matter.) The true reason is more subtle: if a set is both compact and Gδ then it is indeed completely determined by countably many factors. But your "explanation" does not combine these two conditions, thus it cannot be true. --Boris Tsirelson (talk) 12:02, 5 November 2012 (UTC)

Definition
It would be nice to see a source for the definition in the current lead: the sigma-algebra generated by compact $$G_\delta$$ sets (or equivalently, by compactly supported continuous functions), provided that the given space is locally compact. It looks natural; but in the discrete space it gives the countable/cocountable σ-algebra, while Halmos gives the σ-ring of countable sets, and Dudley the σ-algebra of all sets. Boris Tsirelson (talk) 12:27, 29 March 2013 (UTC)

Doubt
In Section "Properties" we see: "Baire sets coincide with Borel sets in Euclidean spaces and all their subsets (treated as topological spaces)." However, it fails for a non-Lebesgue-measurable set. In particular, if a subset of [0,1] is of inner measure 0 and outer measure 1, then all its compact subsets are of measure 0; thus, the σ-algebra generated by these compact sets contains only sets of outer measure 0 or 1; but the Borel σ-algebra (on the given set) contains also sets of outer measure 0.5 (for instance). Boris Tsirelson (talk) 19:00, 4 April 2013 (UTC)

But wait; according to the lead, Baire sets are well-defined only in locally compact spaces. Surely, an arbitrary subset of a Euclidean space is not locally compact. Boris Tsirelson (talk) 19:03, 4 April 2013 (UTC)

Well, under Dudley's definition the problem disappears. The only remaining trouble is that the first part of the lead emphasizes compactness, while Dudley's definition ignores compactness. Boris Tsirelson (talk) 06:48, 5 April 2013 (UTC)

No more stub
Thanks to User:R.e.b. it is not at all a stub now. My objections above are now obsolete. Boris Tsirelson (talk) 14:50, 6 April 2013 (UTC)

The set of rational numbers
I don't understand the concept of a Baire set. On the one hand, it's supposed to be the same as a Borel set on a space like R. The set of rationals is a countable union of closed sets (namely the sets that contain just one rational number), so Q must be a Borel set. But on the other hand, a Baire set is supposed to have a characteristic function which is a "Baire function", defined as "the smallest class of functions containing all continuous real valued functions and closed under pointwise limits of sequences", which is the same as what's called a Baire class 1 function in Baire function. But the characteristic function of Q is not the limit of a sequence of continuous functions (as pointed out in Baire function). So what's the explanation? Eric Kvaalen (talk) 13:37, 13 May 2013 (UTC)
 * No, the Baire class 1 functions are not a class closed under pointwise limits, since their pointwise limits are (generally) Baire class 2 functions. Boris Tsirelson (talk) 13:55, 13 May 2013 (UTC)
 * Ah, I didn't catch that. Thanks. Eric Kvaalen (talk) 15:52, 13 May 2013 (UTC)

Assessment comment
Substituted at 01:47, 5 May 2016 (UTC)