$\textbf{Construction of a measure from an outer measure}$
Let $X$ be a set, let $\mu^*$ be an outer measure on $X$, and let $\mathscr{M}_{\mu^*}$ be the collection of all $\mu^*$-measurable subsets of $X$. Then $\mathscr{M}_{\mu^*}$ is a $\sigma$-algebra, and the restriction of $\mu^*$ to $\mathscr{M}_{\mu^*}$ is a measure on $\mathscr{M}_{\mu^*}$.
$\textbf{Construction of a measure from an outer measure}$
Let $X$ be a set, let $\mu^*$ be an outer measure on $X$, and let $\mathscr{M}_{\mu^*}$ be the collection of all $\mu^*$-measurable subsets of $X$. Then $\mathscr{M}_{\mu^*}$ is a $\sigma$-algebra, and the restriction of $\mu^*$ to $\mathscr{M}_{\mu^*}$ is a measure on $\mathscr{M}_{\mu^*}$.
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๐๐จ๐ง๐ฌ๐ญ๐ซ๐ฎ๐๐ญ๐ข๐จ๐งย ๐จ๐ย ๐ย ๐ฆ๐๐๐ฌ๐ฎ๐ซ๐ย ๐๐ซ๐จ๐ฆย ๐๐งย ๐จ๐ฎ๐ญ๐๐ซย ๐ฆ๐๐๐ฌ๐ฎ๐ซ๐
Let ๐ be a set, let ๐* be an outer measure on ๐, and let โณ๐* be the collection of all ๐*-measurable subsets of ๐. Then โณ๐* is a ๐-algebra, and the restriction of ๐* to โณ๐* is a measure on โณ๐*.