Editing Metric space (section)

=== Metric measure spaces ===

Real analysis makes use of both the metric on <math>\R^n</math> and the [[Lebesgue measure]].  Therefore, generalizations of many ideas from analysis naturally reside in [[metric measure space]]s: spaces that have both a [[measure (mathematics)|measure]] and a metric which are compatible with each other.  Formally, a ''metric measure space'' is a metric space equipped with a [[Borel regular measure]] such that every ball has positive measure.{{sfn|Heinonen|2007|p=191}}  For example Euclidean spaces of dimension {{mvar|n}}, and more generally {{mvar|n}}-dimensional Riemannian manifolds, naturally have the structure of a metric measure space, equipped with the [[Lebesgue measure]].  Certain [[fractal]] metric spaces such as the [[Sierpiński gasket]] can be equipped with the α-dimensional [[Hausdorff measure]] where α is the [[Hausdorff dimension]].  In general, however, a metric space may not have an "obvious" choice of measure.

One application of metric measure spaces is generalizing the notion of [[Ricci curvature]] beyond Riemannian manifolds.  Just as {{math|CAT(''k'')}} and [[Alexandrov space]]s generalize sectional curvature bounds, [[RCD space]]s are a class of metric measure spaces which generalize lower bounds on Ricci curvature.<ref>{{cite journal |last1=Gigli |first1=Nicola |title=Lecture notes on differential calculus on RCD spaces |journal=Publications of the Research Institute for Mathematical Sciences |date=18 October 2018 |volume=54 |issue=4 |pages=855–918 |doi=10.4171/PRIMS/54-4-4 |arxiv=1703.06829|s2cid=119129867 }}</ref>