a powerful tool to study the geometry of radon measures is the decomposability bundle, which i introduced with alberti in [on the differentiability of lipschitz functions with respect to measures in the euclidean space, gafa, 2016]. this is a map which, roughly speaking, captures at almost every point the tangential directions to the lipschitz curves along which the measure can be disintegrated. in this talk i will discuss some recent applications of this flexible tool, including a characterization of rectifiable measures as those measures for which lipschitz functions admit a lusin type approximation with functions of class ${c^1}$, the converse of pansu's theorem on the differentiability of lipschitz functions between carnot groups, and a characterization of federer-fleming flat chains with finite mass.