Two sufficient conditions for rectifiable measures

We identify two sufficient conditions for locally finite Borel measures on ${\mathbb{R}}^n$ to give full mass to a countable family of Lipschitz images of ${\mathbb{R}}^m$. The first condition, extending a prior result of Pajot, is a sufficient test in terms of $L^p$ affine approximability for a locally finite Borel measure $\mu$ on ${\mathbb{R}}^n$ satisfying the global regularity hypothesis $$\underset{r\downarrow 0}{\mathrm{lim\hspace{0.17em}sup}}\mu (B(x,r))\mathrm{/}{r}^m<\mathrm{\infty }\text{at }\mu \text{-a.e. }x\in {\mathbb{R}}^n$$ to be $m$-rectifiable in the sense above. The second condition is an assumption on the growth rate of the 1-density that ensures a locally finite Borel measure $\mu$ on ${\mathbb{R}}^n$ with $$\underset{r\downarrow 0}{\lim }\mu (B(x,r))\mathrm{/}r=\mathrm{\infty }\text{at }\mu \text{-a.e. }x\in {\mathbb{R}}^n$$ is 1-rectifiable.