Lusin-type approximation of Sobolev by Lipschitz functions, in Gaussian and $RCD(K,\mathrm{\infty })$ spaces

We establish new approximation results, in the sense of Lusin, of Sobolev functions by Lipschitz ones, in some classes of non-doubling metric measure structures. Our proof technique relies upon estimates for heat semigroups and applies to Gaussian and $RCD(K,\mathrm{\infty })$ spaces. As a consequence, we obtain quantitative stability for regular Lagrangian flows in Gaussian settings.