6:58 PM
Maybe it will be useful to remind that instructions how to use MathJax in SE chat can be found
here and
here.
I got somewhat interested in the following question:
8
This post is primarily a reference request.
In combinatorics and other areas, we use filter quantifiers to simplify the statements of various definitions, theorems and proofs. The general idea is that, if $\mathcal{F}$ is a filter on a set $W$, and $P(x)$ is a property that an element of $W$ m...
It has two parts: Carl Mummert asks about filter quantifiers and about limits of a sequence along a filter. In case of limits along a filter he asks both for further applications of this notion and for some accessible references.
The reason I became interested in this question is that I have spent some time thinking about this notion. In fact, I have written some notes about this notion (unfinished), which are mentioned in my answer
here.
Let me copy here the comments from the above question. (In order to add some context.)
Is it not the same as the notion of filter convergence in general topology? Let $F$ be a filter on $\mathbb{N}$. Then thinking of the sequence as a function $a:\mathbb{N}\to \mathbb{R}$, we can push forward $F$ to obtain a filter $a_*F$ on $\mathbb{R}$ ($U\in a_*F$ if and only if $a^{-1}[U]\in F$). Then $(a_n)$ $F$-converges to $z$ if and only if $a_*F$ converges to $z$. —
Alex Kruckman Dec 1 '14 at 7:15
@Alex Kruckman: I was not aware of that, actually, so thanks. In the post, I was just hoping to head off throwaway answers about convergence of filters in topology. I had a different personal sense of the motivation for convergence of filters; it would be interesting to know if this was actually it. —
Carl Mummert Dec 2 '14 at 0:44
It seems to me like a nice way of talking about ultraproducts. Take a relational language, with a domain and an ultrafilter of substructures (subsets, in this case), then you can talk about their ultraproduct using filter quantification. —
Asaf Karagila Dec 2 '14 at 4:17
I also think the notation makes for a nice statement of Łoś's theorem, and the properties of filter quantifiers can be seen as parts of the proof of the theorem (e.g. $(\mathcal U\, x)[P(x) \land Q(x)]$ holds if and only if $(\mathcal U\, x)P(x)$ and $(\mathcal U\, x) Q(x)$ hold). I think I have seen the notation more in the setting of combinatorics, perhaps for no good reason. @Asaf Karagila —
Carl Mummert Dec 2 '14 at 12:55
For limits, see also: Agnew/Morse,
Extensions of linear functionals, with applications to limits, integrals, measures, and densities, Annals of Mathematics (2) 39 #1 (January 1938), 20-30; van Douwen,
Finitely additive measures on ${\mathbb N},$ Topology and Its Applications 47 (1992), 223-268; Kostyrko/Salat/Wilczynski, $\cal{I}$-
convergence, Real Analysis Exchange 26 #2 (2000-2001), 669-685; Penot,
Compact nets, filters, and relations, Journal of Mathematical Analysis and Applications 93 #2 (1983), 400-417. [See especially
3. Applications, which begins on p. 406.] —
Dave L. Renfro Dec 8 '14 at 15:56
This type of convergence is used as the basic type of convergence ub some Bourbakis's book, see also Dixmier's General Topology and some other references mentioned in my answer
here. But you are probably interested specifically in filters on $\mathbb N$, Dixmier/Bourbaki deal with filters on an arbitrary set. —
Martin Sleziak Jan 11 at 10:53
Perhaps it is also worth mentioning that several books about set-theory mention limits along an ultrafilter. This
Google Books search returns Hrbacek-Jech and Komjáth-Totik. There are also some books on analysis which take Bourbakists approach and define limit of a function along a filter (base).
This Google Books search returns Brown-Pearcy and Zorich. ... —
Martin Sleziak Jan 11 at 18:03
... Maybe some of these books could be considered undergraduate but, again, they do not devote special attention to the case of filters on $\mathbb N$. (On the other hand, Beardon in his book
Limits - A New Approach to Real Analysis takes a different approach. He defines a limit of a net and other types of limit are special cases of this notion.) —
Martin Sleziak Jan 11 at 18:03
I had the feeling that my comments started to digress from the original question. In my opinion, the questions about references and applications of (ultra)limits could be asked separately from the question about filter quantifiers.
(Well, unless the OP is already satisfied with the applications and references mentioned so far.)
I have decided to post my further comments on this here rather than on the main site. It is not ideal, since messages in chat cannot be edited. But it is at least something.
References for limit of a sequence along an (ultra)filter on $\mathbb N$
As I have already mentioned, I have seen definition of this type of limit in some textbook on set theory. In fact, I have seen this definition for the first time in Balcar-Štìpánek: Teorie Množin (Set Theory, In Czech).
This notion is also defined in Hrbáček-Jech: Introduction to set theory.
References for limit of a function along an (ultra)filter on an arbitrary set
Often it is useful to deal with more general limits, where we work with a filter (or a filter base) on an arbitrary set.
In
this answer I have mentioned Hindman=Strauss and Dixmier.
The approach using filters (filter bases) can be used in analysis to define a limit in unifying way. We defined first a limit along a filter base. Then we can get various kind of limits (one=side, infinite, multivalued) by simply changing the filter base.
I believe that this approach is used in some books by Bourbakists. (I like already mentioned Dixmier's General Topology. This was the only textbook I was able to find where limit superior was defined in this generality.)
So I would not be surprised if some textbooks in mathematical analysis took this approach. (Although it seems rather abstract for the first course in analysis.)
As I mentioned, by googling I found Brown-Pearcy and Zorich.
Applications of limit along (ultra)filter
In number theory, the notion of statistical convergence of a sequence is sometimes useful. This can be considered as a special case of convergence along a filter, see also
my answer here.
Limits along ultrafilters are used more frequently. Roughly said, they can be used in a situations, where you need something like a limit, but you need every bounded real sequence to be convergent. (Or every sequence in some compact space.)
Existence of Banach Limits is one application which is often mentioned. (Balcar-Štěpánek, Komjáth-Totik, Hrbáček-Jech)
Another frequently mentioned application is existence of finitely additive measures extending asymptotic density. (Balcar-Štěpánek, Komjáth-Totik)
As I have already said in one of the comments, finitely additive measures and Banach limits are also mentioned here:
14
I'm looking for some interesting applications of ultrafilters and also everything of interest involving ultrafilters. Do you know some applications or interesting things involving ultrafilters?
I'm at the beginning, so I'd prefer some applications that also a beginner could read.
I have collected some applications also
here. This was written when I was discussing with somebody that one type of arguments can be frequently shown both using nets and using limits along an ultrafilter.
I mention there: Banach limits, Krylov-Bogolyubov theorem, existence of invariant means, relation between Banach density and invariant means.
History of the notion of limit along (ultra)filter
I will mention that this notion seems to be rediscovered several times by various mathematicians.
(This paper is otherwise irrelevant here. I only mention it because the historical comments in the introduction.)
Cartan is most frequently credited as the author of this notion. Although not everybody agrees:
6
(too long for a comment to Pete's answer)
Garrett Birkhoff was my Ph.D. advisor. Let me provide a few remarks
of a historical nature.
From a 25-year-old Garrett Birkhoff we have: Abstract 355, "A new
definition of limit" Bull. Amer. Math. Soc. 41 (1935) 636.
(Received September 5, 1935)
Accor...
Similarly, the origins of the notion of (ultra)filter are not entirely clear.
15
I am writing a paper on some of the applications of ultrafilters, especially the ones on $\mathbb{N}$. I thought that it would be interesting to include some information about how the concept got introduced into mathematics. I would suspect that the earliest applications would be in topology or i...
I will also mention that there is class of filters with the following property:
A sequence $(x_n)$ is $\mathcal F$-convergent to $a$ if and only if there exists $M\in\mathcal F$ such that the subsequence $(x_n)_{n\in M}$ converges to $a$ in the usual sense.
P-filters can be characterized in several equivalent ways. Ultrafilters with this property are called p-points.
The paper by Kostyrko-Šalát-Wiczynski, which was mentioned in Dave L. Renfro's comment, proves exactly this. (But they have a different terminology - they call this property AP. And they work with ideals rather than with filters, which is precisely the dual notion.)
I wonder whether the question about references for $\mathcal F$-limit should be posted as a separate question. (I don't know whether linked questions and comments sufficiently answer the question about applications.) As I have feeling that I have somewhat digressed the topic of your post, I have posted some comments
in chat instead. —
Martin Sleziak 3 mins ago
I've seen a couple of other instances of this, and MJD's post is starting to connect some dots.
In going through review queues I see occasionally a message that "the system has detected" that a post is possibly spam or offensive, and that I should review the item carefully (as if other posts sh...
