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

The Minkowski content measure for the Liouville quantum gravity metric