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 PI spaces 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}\lower3pt\hbox{ $|_{ U_i}$}\ll\mathcal L^{ k_i}$. This class is known to contain ${ \sf RCD}^*(K,N)$ spaces. Part of the work we carry out is that to give a meaning to 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 $x$.
Categories: math.DG