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