Field-measure correspondence in Liouville quantum gravity almost surely commutes with all conformal maps simultaneously

In Liouville quantum gravity (or $2d$-Gaussian multiplicative chaos) one seeks to define a measure ${\mu }^h=e^{\gamma h(z)}dz$ where $h$ is an instance of the Gaussian free field on a planar domain $D$. Since $h$ is a distribution, not a function, one needs a regularization procedure to make this precise: for example, one may let $h_{ϵ}(z)$ be the average value of $h$ on the circle of radius $ϵ$ centered at $z$ (or an analogous average defined using a bump function supported inside that circle) and then write $\mu^h = \lim_{\epsilon \to 0} \epsilon^{\frac{\gamma^2}{2}} e^{\gamma h_\epsilon(z)} dz$. If $\varphi :\tilde{D}\to D$ is a conformal map, one can write $\tilde h = h \circ \phi + Q \log |\phi’|$,where$Q = 2/\gamma + \gamma/2$\mathrm{.}Themeasure$\mu^{\tilde h}$on$\tilde D$isthena\mathrm{.}s\mathrm{.}equivalenttothepullbackvia$\phi^{-1}$ofthemeasure$\mu^h$on$D$. Interestingly, although this a.s.
holds for each $\mathit{\text{given}}$ $\varphi$, nobody has ever proved that it a.s.
holds \textit {simultaneously} for all possible $\varphi$. We will prove that this is indeed the case. This is conceptually important because one frequently defines a $\mathit{\text{quantum}}\mathit{\text{surface}}$ to be an equivalence class of pairs $(D,h)$ (where pairs such as the $(D,h)$ and $(\tilde{D},\tilde{h})$ above are considered equivalent) and it is useful to know that the set of pairs $(D,{\mu }^h)$ obtained from the set of pairs $(D,h)$ in an equivalence class is itself an equivalence class with respect to the usual measure pullback relation.