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I read that a choice set for the rational equivalence relation on a set whose outer measure is positive must necessarily be uncountable. I am having trouble seeing why this is true and was hoping someone could explain this to me. I thought that it would have to have the same cardinality as $\mathbb{Q}$, which is countable.

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Countable sets are of measure zero, therefore they have outer measure zero. Therefore if a set has a positive outer measure, it cannot be countable.

Note that while each equivalence class of $\Bbb{R/Q}$ is countable, there are uncountably many of these classes. Therefore a set choosing a point from each equivalence class must be uncountable.

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