Convergence of Manifolds and Metric Spaces with Boundary

We study sequences of oriented Riemannian manifolds with boundary and, more generally, integral current spaces and metric spaces with boundary. {\color{blue}For a metric space, we define its boundary to be the completion of the space minus the space. We impose conditions on these spaces in order to get Gromov-Hausdorff (GH) subconvergence. By adding some other conditions} we prove theorems demonstrating that the Gromov-Hausdorff (GH) and Sormani-Wenger Intrinsic Flat (SWIF) limits of sequences of such metric spaces agree. Thus in particular the limit spaces we get are countably $\mathcal{H}^n$ rectifiable spaces. From these we derive GH compactness theorems for sequences of Riemannian manifolds with boundary where both the GH and SWIF limits agree. For sequences of Riemannian manifolds with boundary we only require nonnegative Ricci curvature, upper bounds on volume, noncollapsing conditions on the interior of the manifold and diameter controls on the level sets near the boundary.