The following notion of upper Banach density was defined (Definition 2.1(c)) by Hindman and Strauss in their paper '[Density in arbitrary semigroups](https://www.researchgate.net/publication/266702090_Density_in_Arbitrary_Semigroups)':


**Definition:** Let $S$ be a semigroup, let $\mathcal{F}=\{F\_i\}\_{i\in \mathcal{I}}$ be a net in $\mathcal{P}\_f(S)$, and let $A\subseteq S$. Then the upper Banach density of $A$ with respect to $\mathcal{F}$ is $$d\_{\mathcal{F}}^\*(A)=\sup\big\{\alpha: (\forall i\in \mathcal{I})(\exists j>i)(\exists x\in S\cup \{1\})(|A\cap(F\_jx)|\geq \alpha |F\_j|)\big\}.$$


Note that for the standard Følner sequence $\mathcal{F}=\{F\_n\}\_{n\in\mathbb{N}}$, where $F\_n=\{1,2,...n\}$, the corresponding upper Banach density is never additive. My question is 


**Question:** For a fixed semigroup $S$ (we can assume this to be $\mathbb{N}$), does there exists a net $\mathcal{F}$ in $\mathcal{P}\_\mathrm{f}(S)$, so that the corresponding upper Banach density becomes completely additive, i.e. for each $A,B\subseteq S \text{ such that } A\cap B=\emptyset,\,d\_{\mathcal{F}}^\*(A\sqcup B)=d\_{\mathcal{F}}^\*(A)+d\_{\mathcal{F}}^\*(B)$?



Additivity of the upper Banach density
For every family $\mathcal F\subset\mathcal P\_f(\Bbb N) $ both a set $\{1, 3,4, 7,8,9, 13,14,15,16,\dots\} $ and its complement have upper Banach density $d^\*\_{\mathcal F}$ equal to $1$.


