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Is it possible to prove that Lebesgue outer measure on $\mathbb{R}$ is additive on the Borel sigma algebra without constructing the Lebesgue sigma algebra? Every book I have checked first constructs the Lebesgue sigma algebra in some way (either by Caratheodory or by defining the Lebesgue sets to be those that can be approximated (for example) from without by closed sets).

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