Segment inequality and almost rigidity structures for integral Ricci curvature

We will show the Cheeger-Colding segment inequality for manifolds with integral Ricci curvature bound. By using this segment inequality, the almost rigidity structure results for integral Ricci curvature will be derived by a similar method as in cite{CC1}. And the sharp Holder continuity result of cite{CoN} holds in the limit space of manifolds with integral Ricci curvature bound.