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We prove that, for every $\alpha > -1$, the pull-back measure $\phi ({\calA}_\alpha)$ of the measure $d{\cal A}_\alpha (z) = (\alpha + 1) (1 -|z|^2)^\alpha \, d{\cal A} (z)$, where ${\cal A}$ is the normalized areameasure on the unit disk $\D$, by every analytic self-map $\phi \colon \D \to\D$ is not only an $(\alpha + 2)$-Carleson measure, but that the measure of theCarleson windows of size $\eps h$ is controlled by $\eps^{\alpha + 2}$ timesthe measure of the corresponding window of size $h$. This means that theproperty of being an $(\alpha + 2)$-Carleson measure is true at allinfinitesimal scales. We give an application by characterizing the compactnessof composition operators on weighted Bergman-Orlicz spaces.

Infinitesimal Carleson property for weighted measures induced by analytic self-maps of the unit disk