*Submitted Paper*

**Inserted:** 30 nov 2016

**Last Updated:** 13 jul 2018

**Year:** 2016

**Abstract:**

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 $L^2$-sections of the `Gromov-Hausdorff tangent bundle'.

The class of spaces $({\rm X},{\sf d},{\mathfrak m})$ we consider are those that for every $\varepsilon>0$ admit a countable collection of Borel sets $(U_i)$ covering ${\mathfrak m}$-a.e.\ ${\rm X}$ and corresponding $(1+\varepsilon)$-biLipschitz maps $\varphi_i:U_i\to{\mathbb R}^{k_i}$ such that $(\varphi_i)_*({\mathfrak m}\vert_{U_i})\ll\mathcal L^{k_i}$. For technical reasons we shall also require a priori that the Sobolev space $W^{1,2}({\rm X})$ is reflexive (a posteriori such space is proved to be Hilbert). Notice that ${\sf RCD}^*(K,N)$ spaces fit in our framework.

Part of the work we carry out is that to give a meaning to the notion of $L^2$-sections of the Gromov-Hausdorff tangent bundle, in particular explaining what it means to have a measurable map assigning to ${\mathfrak m}$-a.e.\ $x\in {\rm X}$ an element of the pointed-Gromov-Hausdorff limit of the blow-up of ${\rm X}$ at $x$.

With respect to a previous version of the paper, we removed the assumption that the given space is doubling and supports a Poincar\'e inequality.

**Keywords:**
Gromov-Hausdorff tangent space, tangent module

**Download:**