Manifolds with lower Ricci and L1-Sectional curvature control

Suppose that a sequence of Riemannian manifolds with Ricci curvature ≥ -k² converges to a Gromov-Hausdorff limit X. We show that if the amount of sectional curvature below K of the limiting manifolds approaches 0 in a suitable L¹-sense, then X is an Alexandrov space of curvature ≥ K. As applications we present several generalizations of classical theorems.