  A Liouville quantum gravity (LQG) surface is a natural random two-dimensional
surface, initially formulated as a random measure space and later as a random
metric space. We show that the LQG measure can be recovered as the Minkowski
measure with respect to the LQG metric, answering a question of Gwynne and
Miller (arXiv:1905.00383). As a consequence, we prove that the metric structure
of a $\gamma$-LQG surface determines its conformal structure for every $\gamma
\in (0,2)$. Our primary tool is the continuum mating-of-trees theory for
space-filling SLE. In the course of our proof, we also establish a H\"older
continuity result for space-filling SLE with respect to the LQG metric.
