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It's a consequence of Lebesgue's theorem that every measurable $E\subset\mathbb{R}^n$ has a metric density that's $1$ a.e. on $E$ and $0$ a.e. on $\mathbb{R}^n\setminus E$. What are the most general conditions on the measure space for this property to hold?

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up vote 4 down vote accepted

It is true for doubling metric-measure spaces. Check Theorem 1.8 in "Lectures on analysis on metric spaces" by Heinonen.

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