Are the extremal points of a certain set of functions $\mathcal P(\mathbf N) \to \bf R$ weakly additive?

Question

Let an upper density (on $\mathbf N$) be a (set) function $f: \mathcal P(\mathbf N) \to \mathbf R$ such that, for all $X, Y \subseteq \bf N$ and $h,k \in \mathbf N^+$, the following hold:

(F1) $f(\mathbf N) = 1$;

(F2) $f(X) \le f(Y)$ whenever $X \subseteq Y$;

(F3) $f(X \cup Y) \le f(X) + f(Y)$;

(F4) $f(k \cdot X + h) = \frac{1}{k} f(X)$, where $k \cdot X + h := \{kx+h: x \in X\}$.

It was noted in another thread (here) that an upper density $f$ doesn't need to be weakly additive, meaning that $f(X \cup Y) = f(X) + f(Y)$ for all disjoint $X, Y \subseteq \bf N$ such that $Y$ is an (infinite) arithmetic progression (namely, a set of the form $k \cdot \mathbf N + h$ for some $h \in \mathbf N$ and $k \in \mathbf N^+$).

There are, however, a bunch of upper densities that are weakly additive, which is notably true of the upper logarithmic, upper asymptotic, and upper Banach densities (along with at least uncountably many others).

In fact, upper densities form a convex subset, $\mathscr U$, of the real vector space, $\mathcal B(\mathcal P(\mathbf N), \mathbf R)$, of bounded functions $\mathcal P(\mathbf N) \to \bf R$, so it makes sense to ask how the extreme points of $\mathscr{U}$, relative to the linear structure of $\mathcal B(\mathcal P(\mathbf N), \mathbf R)$, look like. Hence my question:

Q. Is it true that every extreme point of $\mathscr{U}$ is a weakly additive upper density? (The answer might depend on the axiom of choice, so let us assume to work in ZFC.)

Here are some positive results: If $\preceq$ denotes the (partial) order on the set, $\hom(\mathcal P(\mathbf N), \mathbf R)$, of all functions $\mathcal P(\mathbf N) \to \bf R$ defined by $f \preceq g$ iff $f(X) \le g(X)$ for all $X \subseteq \bf N$, then it is seen that $\mathscr{U}$ is a complete subsemilattice of the join-semilattice $(\hom(\mathcal P(\mathbf N), \mathbf R), \preceq)$, meaning that, whenever $\mathscr{D}$ is a nonempty subset of $\mathscr{U}$, the set $$\{u \in \mathscr U: f \preceq u\text{ for all }f \in \mathscr{D}\}$$ has a least element, relative to the order $\preceq$, that still belongs to $\mathscr{U}$. In particular, $\mathscr{U}$ has a maximum (again, relative to the order $\preceq$), which is given by the upper Buck density (on $\bf N$), that is the function $$ \mathfrak{b}^\ast: \mathcal P(\mathbf N) \to \mathbf R: X \mapsto \inf_{S \in \mathscr{A}: X \subseteq S} \mathsf{d}^\ast(S), $$ where $\mathscr{A}$ is the set of all subsets of $\mathbf N$ that can be written as a finite union of arithmetic progressions, and $\mathsf d^\ast$ is the upper asymptotic density (on $\bf N$).

It is found that $\mathfrak b^\ast$ is a weakly additive upper density, and on the other hand, it can be proved, under the axiom of choice, that $\mathscr{U}$ has at least uncountably many minimal elements, relative to the order $\preceq$ (I don't know if the existence of even one minimal point can be proved in ZF).

Now, it is not difficult to prove that minimal and maximal elements of $\mathscr{U}$ are extreme points of $\mathscr{U}$; unfortunately, it is not clear to me if also the converse is true (if yes, the answer to question Q would be in the affirmative), though I don't believe so.

Answer 1

I think that the answer is No.

The space $\mathcal B(\mathcal P(\mathbf N), \mathbf R)$ with pointwise convergence is a locally convex space. (It can be considered a subspace of $\mathbf R^{\mathcal P(\mathbf N)}$.)

The set $\mathscr U$ is compact in this space. (It can be identified with a closed subset of $[0,1]^{\mathcal P(\mathbf N)}$.)

So by Krein-Milman theorem we get that $\mathscr U$ is closed convex hull of the set of all extreme points of $\mathscr U$.

On the other hand, the set of all weakly additive upper densities is closed and convex subset of $\mathcal B(\mathcal P(\mathbf N), \mathbf R)$. (It suffices to check that limit of a convergent net of weakly additive upper densities is again a weakly additive upper density and that a convex combination of weakly upper densities is a weakly upper density.) So if all extreme points of $\mathscr U$ were weakly additive, the same would be true for the whole set $\mathscr U$.