We endow each closed, orientable Alexandrov space $(X, d)$ with an integral
current $T$ of weight equal to 1, $\partial T = 0 and \set(T) = X$, in other
words, we prove that $(X, d, T)$ is an integral current space with no boundary.
Combining this result with a result of Li and Perales, we show that
non-collapsing sequences of these spaces with uniform lower curvature and
diameter bounds admit subsequences whose Gromov-Hausdorff and intrinsic flat