New invariants of Gromov–Hausdorff limits of Riemannian surfaces with curvature bounded below

Let { X_i} be a sequence of compact n-dimensional Alexandrov spaces (e.g. Riemannian manifolds) with curvature uniformly bounded below which converges in the Gromov–Hausdorff sense to a compact Alexandrov space X. The paper (Alesker in Arnold Math J 4(1):1–17, 2018) outlined (without a proof) a construction of an integer-valued function on X; this function carries additional geometric information on the sequence such as the limit of intrinsic volumes of the X_i. In this paper we consider sequences of closed 2-surfaces and (1) prove the existence of such a function in this situation; and (2) classify the functions which may arise from the construction.