Completion (4). Inner measure and outer measure. Let \((X, \mathscr{A}, \mu)\) be a finite measure space. Define for every \(E \subset X\) the outer resp. inner measure

\[\begin{aligned}

& \mu^{*}(E):=\inf \{\mu(A): A \in \mathscr{A}, A \supset E\}, \\

& \mu_{*}(E):=\sup \{\mu(A): A \in \mathscr{A}, A \subset E\}

\end{aligned}\]

(i) Show that for all \(E, F \subset X\)

\[\begin{array}{cc}

\mu_{*}(E) \leqslant \mu^{*}(E), & \mu_{*}(E)+\mu^{*}\left(E^{c}ight)=\mu(X), \\

\mu^{*}(E \cup F) \leqslant \mu^{*}(E)+\mu^{*}(F), & \mu_{*}(E)+\mu_{*}(F) \leqslant \mu_{*}(E \cup F) .

\end{array}\]

(ii) For every \(E \subset X\) there exist sets \(E_{*}, E^{*} \in \mathscr{A}\) such that \(\mu\left(E_{*}ight)=\mu_{*}(E)\) and \(\mu\left(E^{*}ight)=\mu^{*}(E)\).

[use the definition of ' \(\infty\) ' to find sets \(E^{n} \supset E\) with \(\mu\left(E^{n}ight)-\mu^{*}(E) \leqslant \frac{1}{n}\) and consider \(\bigcap_{n} E^{n} \in \mathscr{A}\).]

(iii) Show that \(\mathscr{A}^{*}:=\left\{E \subset X: \mu_{*}(E)=\mu^{*}(E)ight\}\) is a \(\sigma\)-algebra and that it is the completion of \(\mathscr{A}\) w.r.t. \(\mu\). Conclude, in particular, that \(\left.\mu^{*}ight|_{\mathscr{A}^{*}}=\left.\mu_{*}ight|_{\mathscr{A}^{*}}=\bar{\mu}\) if \(\bar{\mu}\) is the completion of \(\mu\).