Consider the function $\mu: \mathcal{P}(\mathbb{N}) \rightarrow [0, \infty]$ defined by
$$\mu(A) := \limsup\limits_{n \rightarrow \infty} \frac{1}{n} \cdot \#(A \cap \{1, \dots, n\})$$
Is this function a measure or an outer measure?
It's easy to see that $\mu(\varnothing) = 0$ and I believe that monotonicity also holds. However, I am unsure if the function is also $\sigma$-additive (or $\sigma$-subadditive). I arrived at the point that \begin{align*} \mu\left(\dot{\bigcup_{i \in \mathbb{N}}} A_i \right) &= \limsup\limits_{n \rightarrow \infty} \frac{1}{n} \cdot \#\left(\left(\dot{\bigcup_{i \in \mathbb{N}}} A_i \right) \cap \{1, \dots, n\}\right) \\ &= \limsup\limits_{n \rightarrow \infty} \frac{1}{n} \cdot \sum_{i \in \mathbb{N}}\#(A_i \cap \{1, \dots, n\}) \end{align*}
However, I don't think that I can pull out the infinite sum that easily. On the other hand, I haven't been able to find a counterexample. I would be thankful for any help!