Since I don't wether the definition I have are the standard ones, we define
An outer measure $\mu : \mathcal{P}(X) \longmapsto [0,+\infty]$ every $\sigma$-subadditive map, where given $\Xi \subset \mathcal{P}(X)$, $\mu : \Xi \longmapsto [0,+\infty]$ is said to be subadditive if, $\forall \hspace{0.1cm} S \in \Xi$ and $\forall \left\lbrace F_{n}\right\rbrace_{n \in \mathbb{N}} \subset \Xi$ countable covering such that $\bigcup\limits_{n \in \mathbb{N}} F_{n} \subset \Xi$, $\mu(A) \leq \sum\limits_{n \in \mathbb{N}} \mu(F_{n})$.
I think the $[\Rightarrow]$ is basically using the definition, since if $\left.\alpha^{*}\right|_{\Xi} = \alpha$, the subadditivity of $\alpha^{*}$ guarantees that $\left.\alpha^{*}\right|_{\Xi} = \alpha$ is subadditive on $\Xi$.
I'm stuck with $[\Leftarrow]$, the condition given, the subadditivity on $\Xi$ is clearly necessary, but I don't immediately see how I could well-define $\mu^{*}$ on $\mathcal{P}(X)$ with the property of subbaditivity $\forall S \subset \mathcal{P}(X)$.
Any hint or solution will be appreciated.