Abstract: We address the strong stochastic stability of a broad class of discrete-time dynamical systems - the non-uniformly expanding (NUE) maps - when some random noise is introduced in the deterministic dynamics. In this first part, we provide some motivations and general concepts on NUE maps and random perturbations. We give a tour on the results we are interested and focus our attention in the construction of a random version of induced Gibbs-Markov maps for random NUE maps.