On two notions of curvature on singular surfaces

In this paper, we investigate the equivalence of two distinct notions of curvature bounds on singular surfaces. The first notion involves inequalities of the form $\omega\geq\kappa\mu$ (resp. $\omega\leq\kappa\mu$) where $\omega$ is the curvature measure and $\mu$ the Hausdorff measure. The second notion is the classical Alexandrov curvature bound CBB (resp. CAT). We demonstrate that these two definitions are, in fact, equivalent. Specifically, we fill an important gap in the theory by showing that the inequalities imply the corresponding Alexandrov CBB (resp. CAT) bound. One striking application of our result is that, in combination with a result of Petrunin, the lower bound $\omega\geq\kappa\mu$ implies $RCD(\kappa, 2)$.