abstract: We will discuss a recent result which assert that if \(\mu\) and \(\nu\) are two suitable probability measures in the Euclidean space and \(\phi\) is a good enough bump function, then \( W_2(\phi\mu,\phi\nu) \leq c W_2(\mu,\nu) \). We will describe the application of this result to the so called uniform rectifiability. Some connections of mass transport with singular integrals and rectifiabiity will be also reviewed, if time permits.