  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.
