  Let M be a differentiable manifold. We say that a tensor field g defined on M
is non-regular if g is in some local Lp space or if g is continuous. In this
work we define a mollifier smoothing g_t of g that has the following feature:
If g is a Riemannian metric of class C2, then the Levi-Civita connection and
the Riemannian curvature tensor of g_t converges to the Levi-Civita connection
and to the Riemannian curvature tensor of g respectively as t converges to
zero. Therefore this mollifier smoothing is a good starting point in order to
generalize objects of the classical Riemannian geometry to non-regular
Riemannian manifolds. Finally we give some applications of this mollifier
smoothing. In particular, we generalize the concept of Lipschitz-Killing
curvature measure for some non-regular Riemannian manifolds.
