Equivalence of two different notions of tangent bundle on rectifiable metric measure spaces

We prove that for a suitable class of metric measure spaces the abstract notion of tangent module as defined by the first author can be isometrically identified with the space of L2-sections of the ‘Gromov-Hausdorff tangent bundle’. The key assumption that we make is a form of rectifiability for which the space is ‘almost isometrically’ rectifiable (up to m-null sets) via maps that keep under control the reference measure. We point out that RCD∗(K, N) spaces fit in our framework.