Integral bounds on curvature and Gromov-Hausdorff limits

We develop a new approach to, and small extension of, results of Cheeger, Colding and Tian concerning the $L^{k\mathrm{/}2}$ norm of the curvature of a Riemannian manifold Gromov-Hausdorff close to a codimension $k$ singularity.