The coordinate change formula for the Liouville quantum gravity metric holds for all conformal maps simultaneously

Liouville quantum gravity (LQG) is, heuristically, a theory of random Riemannian geometry with Riemannian metric tensor $e^{\gamma h}(\mathrm{d}{x}^2+\mathrm{d}{y}^2)$, where $h$ is a variant of the Gaussian free field and $\gamma >0$ is a parameter. If $U\subset \mathbb{C}$ is an open set, $\phi \text{\hspace{0.17em}⁣}:U\to \phi (U)$ is a conformal map, and $h^{\phi }=h\circ {\phi }^{-1}+Q\log \mathrm{\mid }({\phi }^{-1}{)}^{\mathrm{\prime }}\mathrm{\mid }$ (where $Q=Q(\gamma )$ is a parameter), then the LQG surface on $U$ defined with field $h$ is equivalent to the LQG surface on $\phi (U)$ with field $h^{\phi }$. This equivalence is meant in the sense that the area measures and distance functions on these surfaces are almost surely equivalent. It is known for the area measure that, in fact, this equivalence holds almost surely for all conformal maps $\phi$ simultaneously (Sheffield-Wang 2016). We prove the corresponding result for the distance function. This makes precise the frequently used heuristic definition that a quantum surface is a random equivalence class of domains equipped with the LQG area measure and LQG distance function.