Wasserstein distance and the rectifiability of doubling measures: part I

Let mu be a doubling measure in R-n. We investigate quantitative relations between the rectifiability of mu and its distance to flat measures. More precisely, for x in the support Sigma of mu and r > 0, we introduce a number alpha(x, r) is an element of (0, 1] that measures, in terms of a variant of the L-1-Wasserstein distance, the minimal distance between the restriction of mu to B(x, r) and a multiple of the Lebesgue measure on an affine subspace that meets B(x, r/2). We show that the set of points of Sigma where integral(1)(0) alpha(x, r) dr/r < infinity can be decomposed into rectifiable pieces of various dimensions. We obtain additional control on the pieces and the size of mu when we assume that some Carleson measure estimates hold.