The question of whether liquid atomization depends on instability dynamics (through refinements of Rayleigh-Plateau, Rayleigh-Taylor, or Kelvin-Helmholtz mechanisms) or on turbulent cascades, as suggested by Richardson and Kolmogorov, is still open. In this paper, experimental results reveal that both mechanisms are needed to explain the probability density functions (PDFs) of the droplets in a spray obtained from an industrial fan spray nozzle. Instability of Rayleigh-Taylor type controls the size of the largest droplets while the smallest droplets follow a PDF given by a turbulent cascading mechanism characterized by a log-Lévy stable law that has a stability parameter equal to 1.70. This value is very close to the inverse value of the Flory exponent and can be related to a recent model developed by N. Rimbert for intermittency modeling stemming from self-avoiding random vortex stretching.