Let $\mu^\star$ be a real-valued function defined on the power set of the positive integers $\mathbf{N}^+$ such that for all $X,Y\subseteq \mathbf{N}^+$ the following axioms hold:

(F1) $\mu^\star(\mathbf{N}^+)=1$;

(F2) $\mu^\star(X) \le \mu^\star(Y)$ if $X\subseteq Y$;

(F3) $\mu^\star(X\cup Y) \le \mu^\star(X)+\mu^\star(Y)$;

(F4) $\mu^\star(\{kx:x \in X\})=\mu^\star(X)/k$ for all $k \in \mathbf{N}^+$;

(F5) $\mu^\star(\{x+h: x \in X\})=\mu^\star(X)$ for all $h \in \mathbf{N}^+$.

A function of this type is said to be an *(arithmetic) upper density*. The set of these functions include the upper asymptotic, Banach, logarithmic, analytic, Polya, Buck densities and many others. Related questions on these type of functions can be found [here][1] and [here][2].

At this point we can defined its associated lower density $\mu_\star$ for all $X\subseteq \mathbf{N}^+$ by 
$$\mu_\star(X)=1-\mu^\star(X^c).$$
Now, all examples I have in mind are superadditive functions, namely
$$
\mu_\star(X\cup Y) \ge \mu_\star(X)+\mu_\star(Y)
$$
whenever $X$ and $Y$ are disjoint subsets of $\mathbf{N}^+$. Does this property hold in general?

After an easy manipulation, the question turns out to be equivalent to the following:
> **Question.** Let $\mu^\star$ be an upper density on $\mathbf{N}^+$, that is, a function satisfying axioms (F1)-(F5). Is it true that if $X,Y$ are subsets of $\mathbf{N}^+$ such that $X\cup Y=\mathbf{N}^+$ then
$$
1+\mu^\star(X\cap Y) \le \mu^\star(X)+\mu^\star(Y)?
$$

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In turn, this can be viewed as a strenghtening of (F3) above; moreover, it would imply that the induced density $\mu$, which can be seen as the restriction of $\mu^\star$ on $\{X\subseteq \mathbf{N}^+:\mu^\star(X)=\mu_\star(X)\}$, is additive.


  [1]: https://mathoverflow.net/questions/214878/additivity-of-upper-densities-with-respect-to-arithmetic-progressions-of-integer
  [2]: https://mathoverflow.net/questions/220268/does-a-monotone-subadditive-f-mathcalp-bf-n-to-0-1-admit-a-finite-part