  We characterize $n$-rectifiable metric measure spaces as those spaces that
admit a countable Borel decomposition so that each piece has positive and
finite $n$-densities and one of the following: is an $n$-dimensional Lipschitz
differentiability space; has $n$-independent Alberti representations; satisfies
David's condition for an $n$-dimensional chart. The key tool is an iterative
grid construction which allows us to show that the image of a ball with a high
density of curves from the Alberti representations under a chart map contains a
large portion of a uniformly large ball and hence satisfies David's condition.
This allows us to apply previously known "biLipschitz pieces" results on the
charts.
