  We establish bounds for the measure of deviation sets associated to
continuous observables with respect to not necessarily invariant weak Gibbs
measures. Under some mild assumptions, we obtain upper and lower bounds for the
measure of deviation sets of some non-uniformly expanding maps, including
quadratic maps and robust multidimensional non-uniformly expanding local
diffeomorphisms. For that purpose, a measure theoretical weak form of
specification is introduced and proved to hold for the robust classes of
multidimensional nonuniformly expanding local diffeomorphisms and Viana maps.
