Given a measure $ν$ on a regular planar domain $D$, the Gaussian multiplicative chaos measure of $ν$ studied in this paper is the random measure ${\widetilde ν}$ obtained as the limit of the exponential of the $γ$-parameter circle averages of the Gaussian free field on $D$ weighted by $ν$. We investigate the dimensional and geometric properties of these random measures. We first show that if $ν$ is a finite Borel measure on $D$ with exact dimension $α>0$, then the associated GMC measure ${\widetilde ν}$ is non-degenerate and is almost surely exact dimensional with dimension $α-\frac{γ^2}{2}$, provided $\frac{γ^2}{2}<α$. We then show that if $ν_t$ is a Hölder-continuously parameterized family of measures then the total mass of ${\widetilde ν}_t$ varies Hölder-continuously with $t$, provided that $γ$ is sufficiently small. As an application we show that if $γ<0.28$, then, almost surely, the orthogonal projections of the $γ$-Liouville quantum gravity measure ${\widetilde μ}$ on a rotund convex domain $D$ in all directions are simultaneously absolutely continuous with respect to Lebesgue measure with Hölder continuous densities. Furthermore, ${\widetilde μ}$ has positive Fourier dimension almost surely.