In this paper we study spaces of holomorphic functions on the right half-plane R, that we denote by M p ω , whose growth conditions are given in terms of a translation invariant measure ω on the closed half-plane R. Such a measure has the form ω = ν ⊗ m, where m is the Lebesgue measure on R and ν is a regular Borel measure on [0, +∞). We call these spaces generalized Hardy–Bergman spaces on the half-plane R. We study in particular the case of ν purely atomic, with point masses on an arithmetic progression on [0, +∞). We obtain a Paley–Wiener theorem for M 2 ω , and consequentely the expression for its reproducing kernel. We study the growth of functions in such space and in particular show that M p ω contains functions of order 1. Moreover, we prove that the orthogonal projection from L p (R,dω) into M p ω is unbounded for p ≠ 2. Furthermore, we compare the spaces M p ω with the classical Hardy and Bergman spaces, and some other Hardy– Bergman-type spaces introduced more recently.