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 $W$ may or may not possess, then $(\forall_\mathcal F\,x) P(x)$ means that $P$ holds on a set of elements that is in $\mathcal F$. This quantifier is also sometimes written $(\mathcal F\, x) P(x)$. There is also a dual quantifier, $(\exists_\mathcal F \, x) P(x)$, which says that the set of elements satisfying $P$ is stationary in $\mathcal F$.
Specific examples include:
If we let $\mathcal F$ be the Fréchet filter of cofinite subsets of $\mathbb N$, we obtain the infinitary quantifiers $\forall^\infty$ and $\exists^\infty$ ("large" sets are cofinite, "small" sets are finite, and stationary sets are infinite).
If we let $\mathcal F$ be the filter of measure 1 subsets of $[0,1]$, we obtain a kind of measure quantifier ("large" sets have measure 1, "small" sets have measure 0, and stationary sets have positive measure).
I am interested in finding any undergraduate textbooks, or any general logic textbooks, that discuss these filter quantifiers in detail. The only references I have been able to locate are graduate-level papers on combinatorics. There are a few isolated internet posts, such as 1 and 2. It would be nice to have something to point a younger student towards.
I am also interested in the following question. Let $\mathcal F$ be the Fréchet filter of cofinite subsets of $\mathbb N$, let $(a_n)$ be a sequence of real numbers and let $z$ be a real number. For each open interval $I$, let $P_I = \{ n \in \mathbb N : a_n \in I\}$. Then the usual definition of convergence can be restated as: $(a_n)$ converges to $z$ if and only if $(\forall_\mathcal F\, x) P_I(x)$ holds for every open interval $I$ containing $z$ (this could also be written $(\forall^\infty\,x)P_I(x)$). And $z$ is a cluster point of $(a_n)$ if and only if $(\exists_\mathcal F\, x) P_I(x)$ for every open interval $I$ containing $z$.
We can generalize the usual notion of convergence by simply replacing the Fréchet filter $\mathbb F$ with any other filter on $\mathbb N$. I am interested in any references about this generalization.
I would suspect there should be a real analysis text that discusses this alternate notion of convergence, at least in exercises. Please note that this is not prima facie the same as the notion of filter convergence in general topology, although comments by Alex Kruckman below show there is a relationship. Since I asked this originally, I've learned from this Tricki post that this method can be used to construct Banach limits, using the method described below (instead of the Hahn-Banach theorem, which is how I had seen it). I would be interested in any other interesting examples of what can be done with this sort of generalized "convergence on a filter". Perhaps there are other Hahn-Banach type results that can be converted to use ultrafilters.