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



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> 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.
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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.



Metric measure spaces: in what sense is analysis on these spaces "non-smooth"
I highly recommend the survey article "Nonsmooth Calculus" by Juha Heinonen, available [here](http://www.ams.org/journals/bull/2007-44-02/S0273-0979-07-01140-8/S0273-0979-07-01140-8.pdf).


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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