  Let $\gamma \in (0,2)$ and let $D_h$ and $\mu_h$ denote the metric and
measure associated with the unit area $\gamma$-Liouville quantum gravity (LQG)
sphere. We show that the closed support of the law of $(D_h,\mu_h)$ includes
all length metrics and probability measures on $\mathbb{S}^2$. That is, if
$\mathfrak d$ is any length metric on the Riemann sphere $\mathbb S^2$ and
$\mathfrak m$ is any probability measure on $\mathbb S^2$ , then with positive
probability $(D_h , \mu_h)$ is close to $(\mathfrak d , \mathfrak m)$ with
respect to the uniform distance and the Prokhorov distance. As an application,
we show that any such metric measure space $(\mathbb S^2 , \mathfrak d ,
\mathfrak m)$ can be approximated with positive probability by appropriately
re-scaled uniform quadrangulations (or triangulations, etc.) in the
Gromov-Hausdorff-Prokhorov sense.
