Second order rectifiability of varifolds of bounded mean curvature

We prove that the support of an $m$ dimensional rectifiable varifold with a uniform lower bound on the density and bounded generalized mean curvature can be covered $\mathscr{H}^{m}$ almost everywhere by a countable union of $m$ dimensional submanifolds of class $\mathcal{C}^{2}$. We obtain this result using the notion of curvature of arbitrary closed sets originally developed in stochastic geometry and extending to our geometric setting techniques developed by Trudinger in the theory of viscosity solutions of PDE's.