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If $\mu_0$ is an outer measure on an algebra, we can extend the premeasure to an outer measure $\mu^*$. By Caratheodory's theorem, the collection of $\mu^*$-measurable sets is a $\sigma$-algebra. Is an outer measure $\mu^*$ restricted to $\mu^*$-measurable sets a measure?

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Yes, it is. That's also part of Caratheodory's extension theorem.

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