  We consider extensions of quasiconformal maps and the uniformization theorem
to the setting of metric spaces $X$ homeomorphic to $\mathbb R^2$. Given a
measure $\mu$ on such a space, we introduce $\mu$-quasiconformal maps $f:X \to
\mathbb R^2$, whose definition involves deforming lengths of curves by $\mu$.
We show that if $\mu$ is an infinitesimally metric measure, i.e., it satisfies
an infinitesimal version of the metric doubling measure condition of David and
Semmes, then such a $\mu$-quasiconformal map exists. We apply this result to
give a characterization of the metric spaces admitting an infinitesimally
quasisymmetric parametrization.
