On 1-regular and 1-uniform metric measure spaces

A metric measure space $(X,\mu )$ is 1-regular if $$0<\underset{r\to 0}{\lim }\frac{\mu (B(x,r))}{r}<\mathrm{\infty }$$ for $\mu$-a.e $x\in X$. We give a complete geometric characterisation of the rectifiable and purely unrectifiable part of a 1-regular measure in terms of its tangent spaces. A special instance of a 1-regular metric measure space is a 1-uniform space $(Y,\nu )$, which satisfies $\nu (B(y,r))=r$ for all $y\in Y$ and $r>0$. We prove that there are exactly three 1-uniform metric measure spaces.