Given a measure on a regular planar domain, the Gaussian multiplicative chaos measure or the Liouville quantum version of it is the random measure obtained as the weak limit of exponential of circle averages of the Gaussian free field weighted by the original measure. We investigate some dimensional and geometric properties of these random measure. We show that if the original measure is exact-dimensional then so is the random measure. We also show that when the dimension of the random measure is large enough, its orthogonal projections to one-dimensional subspaces are absolutely continuous w.r.t Lebesgue measure in every direction, and it has positive Fourier dimension.
