Let $G$ be an amenable group (so locally compact Hausdorff) and also assume it is second countable if needed. My question is that what are the standard ways (across literature) of defining the upper density for a subset of $G$?
If we are talking about $\mathbb{N}$, then given a Folner sequence $\mathcal{F}=\{F_n: n\in \omega\}$ (basically each $F_n$ is finite and for any $m\in \mathbb{N}$, $\lim_{n\to \infty} \frac{|(m+F_n)\Delta F_n|}{|F_n|}=0$), then the upper density associated with $\mathcal{F}$ can be $\bar{d}_\mathcal{F}(A)=\limsup_{n\to \infty} \frac{|A\cap F_n|}{|F_n|}$.
One way to define the density is to replace the counting measure by the Haar measure $\mu$ on $G$ and the Folner sequence in a more general sense ($F_n$ are compact sets now) and we can say $\bar{d}_\mathcal{F}(A)=\limsup_{n\to \infty} \frac{\mu^*(A\cap F_n)}{\mu(F_n)}$, where $\mu^*$ is the outer measure (I want to define density for any subset $A$).
Another way is to say $\bar{d}(A)=\sup\{\alpha: \text{for every finite }H\subset G \text{ and $\epsilon>0$, there is a finite $K$ with }\frac{|hK\Delta K|}{|K|}<\epsilon \forall h\in H \text{ and }\frac{|A\cap K|}{|K|}\geq \alpha\}$$\bar{d}(A)=\sup\{\alpha: \text{for every finite }H\subset G \text{ and $\varepsilon>0$, there is a finite $K$ with }\frac{|hK\Delta K|}{|K|}<\varepsilon\; \forall h\in H \text{ and }\frac{|A\cap K|}{|K|}\geq \alpha\}$.
I'm not a group theorist nor topologist and hopefully this question is okay here. Thanks.
