Isometry vs. measure preserving? Consider functions between two measured metric spaces. What is the relation between an isometry and a function which preserves the measure of subsets? 
This question arose in my head as I thought about the surjection of the Cantor set onto $[0,1]$. 
This question had something, but I don't think it completely settles it.
 A: Isometries preserve the metric space structure. Measure space structure is generally an  entirely different thing;  preserving one thing is unrelated to preserving the other. However, there is an exception: Hausdorff measures, which are defined solely in terms of the metric and therefore behave well under isometries. Precise statement: if $f:X\to Y$ is a bijection between metric spaces such that $d_Y(f(a),f(b))=d_X(a,b)$ for all $a,b\in X$, then the pushforward of Hausdorff measure $\mathcal{H}^d$ on $X$ under $f$ is the Hausdorff measure $\mathcal{H}^d$ on $Y$.
By the way, the counting measure can be understood as the $0$-dimensional Hausdorff measure, so it is included here.  Of course, the counting measure behaves   well under any bijection. 
I could not come up with a natural situation  in which measure-preserving maps are automatically metric-preserving. The problem is that it's hard to construct a metric from a measure without involving additional structures, such as topology. E.g.,  one could try to construct a metric $d_\mu$ from measure $\mu$ by letting
$$d_\mu(a,b)=\inf\{\mu(E): a,b\in E,\ \text{ $E$ is connected}\}$$
but this involves the topological structure too. The above is a sensible construction on the real line; e.g., you   get the standard metric from the Lebesgue measure in this way.
