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.