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In this paper, we study regular sets in metric measure spaces with boundedRicci curvature. We prove that the existence of a point in the regular set ofthe highest dimension implies the positivity of the measure of such regularset. Also we define thee dimension of metric measure spaces and prove the lowersemicontinuity of that under the Gromov-Hausdorff convergence.

A sufficient condition to a regular set of positive measure on RCD spaces