The two maps are defined by: $P( \mathbb N) \rightarrow [0, \infty]$.
1.) $\mu_1(A)$ is $0$ if A is empty and $\infty$ if A is not empty. This is not an outer measure, because the sigma subadditivity is not given. Right?
2.) $\mu_2(A) = $ lim sup $\frac{1}{n} $#$(A \cap \{1, ..., n \})$ is an outer measure. Now I should indicate the system of measurable sets. But what is meant by "system of measurable sets"?