Characterising rectifiable metric spaces using tangent spaces

We characterise rectifiable subsets of a complete metric space $X$ in terms local approximation, with respect to the Gromov--Hausdorff distance, by an $n$-dimensional Banach space. In fact, if $E\subset X$ $\mathcal{H}^n(E)<\infty$ and has positive lower density almost everywhere, we prove that it is sufficient that, at every point each sufficiently small scale, $E$ approximated bi-Lipschitz image Euclidean also introduce generalisation Preiss's tangent measures suitable for setting arbitrary spaces formulate our characterisation measures. This definition equivalent Preiss when ambient Euclidean, measured measure doubling.