Metric measure spaces: in what sense is analysis on these spaces "non-smooth"

I understand the basic definition of a metric measure space to be the following:

A metric measure space is a triple of a space $$X$$, metric $$d$$, and measure $$m$$: $$(X,d,m)$$ in the sense that the metric induces a topology and the measure is the borel measure arising from the sigma field induced by the metric.

I frequently hear/read that, that analysis on these spaces is, informally, analysis in spaces with no a priori smooth structure.

In what sense, formal or informal, is analysis on these spaces "non-smooth"? Does it have to do with the fact the measure isn't complete or something?

Also feel free to change the tags on this question, I'm not sure where it goes.

• It simply has to do with the fact there is no notion of a smooth or differentiable function between metric measure spaces. Jan 28 '19 at 14:28
• One example that comes to mind is the class of Alexandrov spaces, which can be thought of as limits of smooth manifolds. For example the distance function is never smooth on a compact smooth manifold, I think. Jan 28 '19 at 14:39
• Another example is analysis on Cantor sets. No smooth structures. Since you are at Stony Brook you should talk to Raanan Schul or Silvia Ghinassi. They would provide you many interesting examples. Jan 28 '19 at 14:55
• I would compare to the analysis on the "smoothest" spaces, i.e. real vector spaces with an inner product.
– Dirk
Jan 28 '19 at 19:59
• @Wojowu Actually in many situation there are "differentiable" mappings between metric spaces. In particular Cheeger proved existence of a measurable differentiable structure on spaces supporting Poincare inequalities. Mar 5 '20 at 18:05