  Let $A^p_\omega$ denote the Bergman space in the unit disc induced by a
radial weight~$\omega$ with the doubling property $\int_{r}^1\omega(s)\,ds\le
C\int_{\frac{1+r}{2}}^1\omega(s)\,ds$. The positive Borel measures such that
the differentiation operator of order $n\in\mathbb{N}\cup\{0\}$ is bounded from
$A^p_\omega$ into $L^q(\mu)$ are characterized in terms of geometric conditions
when $0<p,q<\infty$. En route to the proof a theory of tent spaces for weighted
Bergman spaces is built.
