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

Abstract

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

arXiv comment15 pages.arXiv admin note: text overlap with arXiv:1603.04162

References in corpus (5)

• Metric measure spaces with Riemannian Ricci curvature bounded from below
• The splitting theorem in non-smooth context
• Behaviour of the reference measure on $\mathsf{RCD}$ spaces under charts
• Orthogonal symmetric non-negative matrix factorization under the stochastic block model
• On dimensions of tangent cones in limit spaces with lower Ricci curvature bounds

Cited by in corpus (3)

• Collapsed Ricci limit spaces as non-collapsed $RCD$ spaces
• Sobolev mappings between RCD spaces and applications to harmonic maps: a heat kernel approach
• Isometric immersions of RCD spaces