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We develop a new approach to, and somewhat extend, results of Cheeger, Colding and Tian ( Geom. Funct. Anal . 12 (2002) 873–914), which give restrictions on the singular sets that can arise as Gromov‐Hausdorff limits in the presence of integral bounds on the curvature. The foundation of this approach is provided by a rigidity result for flat singular spaces, which has independent interest.

Integral bounds on curvature and Gromov–Hausdorff limits