  Let $L$ be a holomorphic line bundle over a compact K\"ahler manifold $X$
endowed with a singular Hermitian metric $h$ with curvature current
$c_1(L,h)\geq0$. In certain cases when the wedge product $c_1(L,h)^k$ is a well
defined current for some positive integer $k\leq\dim X$, we prove that
$c_1(L,h)^k$ can be approximated by averages of currents of integration over
the common zero sets of $k$-tuples of holomorphic sections over $X$ of the high
powers $L^p:=L^{\otimes p}$.
  In the second part of the paper we study the convergence of the Fubini-Study
currents and the equidistribution of zeros of $L^2$-holomorphic sections of the
adjoint bundles $L^p\otimes K_X$, where $L$ is a holomorphic line bundle over a
complex manifold $X$ endowed with a singular Hermitian metric $h$ with positive
curvature current. As an application, we obtain an approximation theorem for
the current $c_1(L,h)^k$ using currents of integration over the common zero
sets of $k$-tuples of sections of $L^p\otimes K_X$.
