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

