Carathéodory's Theorem: the Measurable Sets of an Outer Measure Form a σ-Algebra on Which It Is a Measure

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $X$ be a set, let $\mathcal{P}(X)$ be its power set, let $\mu^{*}$ be an outer measure on $X$, and let $\mathcal{M}=\mathcal{M}(\mu^{*})$ be the set of $\mu^{*}$-measurable subsets of $X$, as in Carathéodory Measurable Sets of an Outer Measure (ii). Let $\infty$, $[0,\infty]$, the order $\le$ and the relation $<$ be as in The Extended Real Line, Its Order, and [0,∞] (i), The Extended Real Line, Its Order, and [0,∞] (ii) and The Extended Real Line, Its Order, and [0,∞] (iii), let $+$ be the sum on $[0,\infty]$, and let $A\cap E$ and $A\setminus E$ be the intersection and difference of sets $A$ and $E$.

(i) A subset $E$ of $X$ lies in $\mathcal{M}$ if and only if $\mu^{*}(A\cap E)+\mu^{*}(A\setminus E)\le\mu^{*}(A)$ for every subset $A$ of $X$ with $\mu^{*}(A)<\infty$.

(ii) $\mathcal{M}$ is a $\sigma$-algebra on $X$; thus $(X,\mathcal{M})$ is a measurable space.

(iii) Every subset $N$ of $X$ with $\mu^{*}(N)=0$ lies in $\mathcal{M}$.

(iv) Since $\mathcal{M}\subseteq\mathcal{P}(X)$ by Carathéodory Measurable Sets of an Outer Measure (ii), the restriction $\mu^{*}|_{\mathcal{M}}$ of $\mu^{*}$ to $\mathcal{M}$ is a map $\mu^{*}|_{\mathcal{M}}:\mathcal{M}\to[0,\infty]$ with $(\mu^{*}|_{\mathcal{M}})(E)=\mu^{*}(E)$ for every $E\in\mathcal{M}$, by Domains and Restrictions of Maps (ii) and Domains and Restrictions of Maps (iii). By (ii), $(X,\mathcal{M})$ is a measurable space, and $\mu^{*}|_{\mathcal{M}}$ is a measure on it.