  Any Borel probability measure supported on a Cantor set of zero Lebesgue
measure on the real line possesses a discrete inverse measure. We study the
validity of the multifractal formalism for the inverse measures of random weak
Gibbs measures supported on the attractor associated with some $C^1$ random
dynamics encoded by a random subshift of finite type, and expanding in the
mean. The study requires, in particular, to develop in this context of random
dynamics a suitable extension of the results known for heterogeneous ubiquity
associated with deterministic Gibbs measures.
