Characterizations of rectifiable metric measure spaces

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 modified versions of previously known ‘biLipschitz pieces' results on the charts.