  We give the definition of $L^p$-convergence of tensor fields with respect to
the Gromov-Hausdorff topology and several fundamental properties of the
convergence. We apply this to establish a Bochner-type inequality which keeps
the term of Hessian on the Gromov-Hausdorff limit space of a sequence of
Riemannian manifolds with a lower Ricci curvature bound and to give a geometric
explicit formula for the Dirichlet Laplacian on a limit space defined by
Cheeger-Colding. We also prove a continuity of the first eigenvalues of the
p-Laplacian with respect to the Gromov-Hausdorff topology.
