Determine the outer measure induced by a given set function (which counts the number of integers in a subset of $\mathbb{R}$)

Question

On the collection $\mathcal{S}$ of all subsets of $\mathbb{R}$, define the set function $\mu: \mathcal{S} \rightarrow \mathbb{R}$ by setting $\mu(A)$ to be the number of integers in $A$. Determine the outer measure induced by $\mu$ and the $\sigma$-algebra of measurable sets.

I was wondering if the following is true: any $E$ is a subset of $\mathbb{R}$ (i.e., $E \in \mathcal{S}$), which implies that $\mu^*(E)=\mu(E)$. For any set $A$, $\mu^*(A)=\mu^*(A \cap E)+\mu^*(A \cap E^{\complement})$ always holds.

Answer 1

$\mu =\sum_{n \in \mathbb Z} \delta_n$ is a measure so the outer measure induced by it $\mu$ itself.

For any sequence of measures $\mu_n$ the sum $\sum_n \mu_n$ is countably additive.