The following notion of Upperupper Banach density was defined (Definition 2.1(c)) by Hindman and Strauss in their paper 'Density in Arbitrary Semigroupsarbitrary 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 $$d_{\mathcal{F}}^*(A)=sup\{\alpha: (\forall i\in \mathcal{I})(\exists j>i)(\exists x\in S\cup \{1\})(|A\cap(F_jx)|\geq \alpha |F_j|)\}.$$$$d_{\mathcal{F}}^*(A)=\sup\big\{\alpha: (\forall i\in \mathcal{I})(\exists j>i)(\exists x\in S\cup \{1\})(|A\cap(F_jx)|\geq \alpha |F_j|)\big\}.$$
Note that for the standard Følner sequence $\mathcal{F}=\{F_n\}_{n\in\mathbb{N}}$, where $F_n=\{1,2,...n\}$, the corresponding Upperupper 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}_f(S)$$\mathcal{P}_\mathrm{f}(S)$, so that the corresponding Upperupper Banach density becomes completely additive, i.e. for each $A,B\subseteq S \text{ such that } A\cap B=\emptyset,\,d_{\mathcal{F}}^*(A\sqcup B)=d_{\mathcal{F}}^*(A)+d_{\mathcal{F}}^*(B)$?
