  This paper is devoted to the study of the weighted Bergman space $A_\omega^p
$ in the unit ball $\mathbb{B}$ of $\mathbb{C}^n$ with doubling weight $\omega$
satisfying $$\int_r^1\omega(t)dt <C \int_{\frac{1+r}{2}}^1\omega(t)dt ,\,\,
0\leq r<1.$$
  The $q-$Carleson measures for $A_\omega^p$ are characterized in terms of a
neat geometric condition involving Carleson block. Some equivalent
characterizations for $A_\omega^p$ are obtained by using the radial derivative
and admissible approach regions. The boundedness and compactness of Volterra
integral operator $T_g:A_\omega^p\to A_\omega^q$ are also investigated in this
paper with $0<p\leq q<\infty$, where $$T_gf(z)=\int_0^1 f(tz)\Re
g(tz)\frac{dt}{t}, ~~\qquad~~~~f\in H(\mathbb{B}), ~~z\in \mathbb{B}. $$
