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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 in 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.

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    $\begingroup$ It simply has to do with the fact there is no notion of a smooth or differentiable function between metric measure spaces. $\endgroup$ – Wojowu Jan 28 at 14:28
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    $\begingroup$ 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. $\endgroup$ – David Hughes Jan 28 at 14:39
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    $\begingroup$ 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. $\endgroup$ – Piotr Hajlasz Jan 28 at 14:55
  • $\begingroup$ I would compare to the analysis on the "smoothest" spaces, i.e. real vector spaces with an inner product. $\endgroup$ – Dirk Jan 28 at 19:59
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I highly recommend the survey article "Nonsmooth Calculus" by Juha Heinonen, available here.

The beginning of the introduction reads:

"The word nonsmooth in the title refers both to functions and spaces. Calculus is a field of study where infinitesimal data yields global information. Mathematicians have been practicing calculus with nonsmooth functions for over a century, but only recently in spaces that are not smooth in the traditional sense. In this article, we first survey calculus with nonsmooth functions and then move on to discuss current advances involving singular spaces."

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