Negative moments for Gaussian multiplicative chaos on fractal sets

The objective of this note is to study the probability that the total mass of a sub-critical Gaussian multiplicative chaos (GMC) with arbitrary base measure $\sigma$ is small. When $\sigma$ has some continuous density w.r.t Lebesgue measure, a scaling argument shows that the logarithm of the total GMC mass is sub-Gaussian near $-\mathrm{\infty }$. However, when $\sigma$ has no scaling properties, the situation is much less clear. In this paper, we prove that for any base measure $\sigma$, the total GMC mass has negative moments of all orders.