Generalization of natural density on $\omega_1$ The natural density (or asymptotic density) of a subset $A$ of $\mathbb{N}$ is defined as $$d(A) = \lim_{n \rightarrow +\infty} \frac{|A\cap \{0,1,\dots, n-1\}|}{n}$$ when such limit exists.
I was wandering if there exists a generalization of such concept that captures a sort of asymptotic density of subsets of larger cardinals. For example a measure (not necessarily in the measure theoretic sense) that extends the natural density and that discriminates between cofinal and bounded subsets of $\omega_1$. Is there? Any idea?
Thanks!
EDIT: I'll try to frame the question in a more specific and different way. What I'm looking for is an additive measure $\mu: \Sigma\subseteq\mathcal{P}(\omega_1) \rightarrow X$ with $(X,\le,+,0)$ being a ordered divisible group, and $\Sigma$ a "reasonable" algebra of the power set (i.e. it includes non-pathological subsets of $\omega_1$, like in the $\omega$ case when it included sets having a definite asymptotic density)  s.t.

*

*$\mu(\omega_1) = 1$, where $1$ is an element of the group strictly greater than $0$

*$\mu(\emptyset) = 0$

*$A\subseteq B \Rightarrow \mu(A) \le \mu(B)$

*$A \cap B = \emptyset \Rightarrow \mu(A \cup B) = \mu(A)+\mu(B)$

*$\mu(n\omega_1) = \frac{1}{n}$, where $n\omega_1$ is the set of all the ordinals less than $\omega_1$ having finite part divisible by $n$

*Given $A,B \subseteq \omega_1$ with $A$ bounded and $B$ cofinal, then $\mu(A) < \mu(B)$
 A: This isn't going to be a complete answer, but just some thoughts that may help you go in the right direction.
First, here are two small points about your question:

*

*Like bof mentioned, natural density on $\omega$ allows cofinal subsets to have density $0$.
So, it might be more reasonable to change your condition (6) to $\mu(A) = 0$ for any bounded $A \subseteq \omega_1$.


*You said $\Sigma \subseteq \mathcal{P}(\omega_1)$ should be an algebra, but in fact this is not even true for natural density.
It's a fun exercise to try to find $A, B \subseteq \omega$, both having natural density, for which neither $A \cup B$ nor $A \cap B$ have natural density.
With that out of the way, the more important point here is that if you want to analogize with natural density as closely as possible you probably want to think about Følner sequences.
If $G$ is a countable left-cancellative semigroup, a (left) Følner sequence of $G$ is a sequence $\Phi := (\Phi_n)_{n \in \mathbb{N}}$ of finite subsets of $G$ satisfying
$$
\lim_{n \to \infty} \frac{|\Phi_n \cap g\Phi_n|}{|\Phi_n|} = 1
$$
for every $g \in G$.
Then, given $A \subseteq G$, we define the density of $A$ with respect to $\Phi$ as
$$
\operatorname{d}_\Phi(A) := \lim_{n \to \infty} \frac{|A \cap \Phi_n|}{|\Phi_n|}.
$$
Then many of the familiar properties of natural density hold for $\operatorname{d}_\Phi$.
In particular, your properties (1)-(4) are immediate from the definition, and you get some other nice properties as well, such as left-translation invariance.
Of course, in your case, the semigroup $(\omega_1, +)$ is not countable, which means that one has to be a bit more careful.
It is fortunately left-cancellative.
Because of this, it may be useful to first try finding “natural” Følner sequences for countable ordinals, such as $(\omega^2, +)$.
This doesn't seem too difficult to do, and might give some ideas for how to approach the uncountable case.
In the case of $(\omega_1, +)$, it seems that no Følner sequence, under the definition I gave, can exist.
Indeed, if $(F_n)$ is any sequence of finite subsets of $\omega_1$, then $\cup F_n$ is finite and so there is some $\alpha \in \omega_1$ larger than any element of this set.
From there it is immediate that the Følner condition will fail with $g = \alpha$.
An idea which might be worth pursuing would be to consider Følner sequences of uncountable length or even Følner nets.
A good reference is probably the paper “Density in arbitrary semigroups” by Neil Hindman and Dona Strauss.
