We prove that, for every α>−1, the pull-back measure ϕ(Aα) of the measure dAα(z)=(α+1)(1−∣z∣2)αdA(z), where A is the normalized area
measure on the unit disk \D, by every analytic self-map \phi \colon \D \to
\D is not only an (α+2)-Carleson measure, but that the measure of the
Carleson windows of size \eps h is controlled by \eps^{\alpha + 2} times
the measure of the corresponding window of size h. This means that the
property of being an (α+2)-Carleson measure is true at all
infinitesimal scales. We give an application by characterizing the compactness
of composition operators on weighted Bergman-Orlicz spaces