Limits of manifolds with a Kato bound on the Ricci curvature

We study the structure of Gromov-Hausdorff limits of sequences of Riemannian manifolds f.M n ; g /g 2A whose Ricci curvature satisfies a uniform Kato bound.We first obtain Mosco convergence of the Dirichlet energies to the Cheeger energy, and show that tangent cones of such limits satisfy the RCD.0; n/ condition.Under a noncollapsing assumption, we introduce a new family of monotone quantities, which allows us to prove that tangent cones are also metric cones.We then show the existence of a well-defined stratification in terms of splittings of tangent cones.We finally prove volume convergence to the Hausdorff n-measure.53C21; 53C23, 31C25, 58J35