In this paper we introduce the family of spaces RM(p,q), 1≤p,q≤+∞. They are spaces of holomorphic functions in the unit disc with average radial integrability. This family contains the classical Hardy spaces (when p=∞) and Bergman spaces (when p=q). We characterize the inclusion between RM(p1,q1) and RM(p2,q2) depending on the parameters. For 1<p,q<∞, our main result provides a characterization of the dual spaces of RM(p,q) by means of the boundedness of the Bergman projection. We show that RM(p,q) is separable if and only if q<+∞. In fact, we provide a method to build isomorphic copies of ℓ∞ in RM(p,∞).