I read a research paper stating we can extend the definition of a Density, that uses a Folner Sequence of countable sets, to one that uses Folner Nets of Uncountable sets (ex: $\mathbb{R}$). I couldn’t decipher the definitions so I'm unsure how to apply this Density to any subset of $\mathbb{R}$ or check a "modified version" of this density equals the Lebesgue measure of these sets.
Let $S$ be a semigroup and let $\mathcal{F}=\langle F_{n} \rangle_{n\in D}$ be a net in $\mathcal{P}_{f}(S)$, where $\mathcal{P}_{f}(S)$ is the set of all non-empty finite subsets of $S$. Then $\mathcal{F}$ is the left Folner Net if and only if for each $s\in S$, the net
$$\left\langle \frac{\left|sF_{n}\triangle F_n\right|}{\left|F_n\right|} \right\rangle_{n\in D}$$
converges to zero. Also $\mathcal{F}$ is a right Folner Net if and only if for each $s\in S$, the net
$$\left\langle \frac{\left|F_{n}s\triangle F_n\right|}{\left|F_n\right|} \right\rangle_{n\in D}$$
converges to zero.
The "modified" Folner Net is simply $\mathcal{F}_{[a,b]}=\langle F_n \cap [a,b] \rangle_{n\in D}$ for $a,b\in\mathbb{R}$
If this is the case, the "modified" density, ${{d}^{*}}_{\mathcal{F}_{[a,b]}}(A\cap[a,b])$, using nets in $\mathcal{P}_{f}(s)$, for $a\le s \le b$ and $A\cap[a,b]\subseteq{S}\cap[a,b]$, is
$$\underline{d}_{\mathcal{F}_{[a,b]}}(A\cap[a,b])\le {{d}^{*}}_{\mathcal{F}_{[a,b]}}(A\cap[a,b]) \le \overline{d}_{\mathcal{F}_{[a,b]}}(A\cap[a,b])$$
(a) The lower density is $$\underline{d}_{\mathcal{F}_{[a,b]}}(A\cap[a,b])=\sup\left\{\alpha:(\exists m \in D)(\forall n \ge m)(\left|A\cap F_n\cap[a,b]\right|)\ge \alpha*\left|F_n\cap[a,b]\right|\right\}$$
(b) The upper density is $$\overline{d}_{\mathcal{F}_{[a,b]}}(A\cap[a,b])=\sup\left\{\alpha:(\forall m \in D)(\exists n \ge m)(\left|A\cap F_n\cap[a,b]\right|)\ge \alpha*\left|F_n\cap[a,b]\right|\right\}$$
(c) And the Density squeezed between is $${{d}^{*}}_{\mathcal{F}_{[a,b]}}(A\cap[a,b])=\sup\left\{\alpha:(\forall m \in D)(\exists n \ge m)(\exists x \in S \cup {1})(\left|A\cap F_n x \cap [a,b]\right|)\ge \alpha*\left|F_n\cap[a,b]\right|\right\}$$
If $A\cap[a,b]=\mathbb{R}\setminus\mathbb{Q}\cap[a,b]$ and $S=[a,b]$ what is ${{d}^{*}}_{\mathcal{F}_{[a,b]}}(A\cap[a,b])$? What if $A=\mathbb{Q}\cap[a,b]$ and $S=[a,b]$? If there is no single value for these densities can we find a Folner Net of $S\cap[a,b]$ where ${{d}^{*}}_{\mathcal{F}_{[a,b]}}(A\cap[a,b])$ give similar results to $\mu(A\cap[a,b])$, where $\mu$ is the Lebesgue Measure?
Edit: If $S\cap[a,b]=[a,b]$ the paper states that ${d^{*}}_{\mathcal{F}_{[a,b]}}(A)\ge\mu(A\cap[a,b])$ where $\mu$ is a countable additive measure. If the measure was Lebesgue, could we take a specific $F_n$ so that ${d^{*}}_{\mathcal{F}_{[a,b]}}(A\cap[a,b])=\mu(A\cap[a,b])$?
Best Answer
Too long for a comment.
A density of a set can be defined by means of any net in $\mathcal P_f(S)$ (see Definition 2.1 of the paper). But I guess that from the amenability point of view a purpose of Følner nets is to define a left-shift invariant measure on the family of subsets of $S$, see, for instance, paragraph after Definition 1.6, Theorems 4.5 and 4.17. For this purpose translations are in the definitions of Følner sequences or nets.
On the other hand, if we a looking for translation invariance then it seems the following (I didn’t perfectly check this paragraph). From the measure point of view we can identify an interval $[0,1)$ with the unit circle $\Bbb T=\{ z\in\Bbb C: |z|=1\}$ which is a group by a map $f:[0,1)\to\Bbb T$, $t\mapsto e^{2\pi it}$. Since $\Bbb T$ is a compact topological group, in admits a Haar measure $\lambda$. Then for any Borel subset $A$ of $[0,1)$ the set $f(A)$ is Borel in $\Bbb T$, so the value $\lambda(f(A))$ is defined and equals to Lebesgue measure of the set $A$.
Recall that by a Haar measure on a locally compact group we understand any left-invariant Borel regular $\sigma$-additive measure that takes positive finite values on compact sets with non-empty interior. It is well-known that each locally compact topological group has a Haar measure and such measure is unique up to a positive multiplicative constant. On the other hand, the Haar measure does not exists in non-locally compact groups.
At last, I remember that recently my boss Taras Banakh considered relations between measures and densities on groups, so maybe in the following papers you can find something useful for you: