  In this paper, using a new technique from harmonic analysis called sparse
domination, we characterize the positive Borel measures including forward,
vanishing, and reverse Bergman Carleson measures. The main novelty of this
paper is determining the reverse Bergman Carleson measures which have remained
open from the work of Luecking [Am. J. Math. 107 (1985) 85-111]. Moreover, in
the case of forward and vanishing measures, our results extend the results of
[J. Funct. Anal. 280 (2021), no. 6, 108897, 26 pp] from $1\leq p\leq q< 2p$ to
all $0<p\leq q<\infty$. In a more general case, we characterize the positive
Borel measures $\mu$ on $\mathbb{B}$ so that the radial differentiation
operator $R^{k}:A_\omega^p(\mathbb{B})\rightarrow L^q(\mathbb{B},\mu)$ is
bounded and compact. Although we consider the weighted Bergman spaces induced
by two-side doubling weights, the results are new even on classical weighted
Bergman spaces.
