  K. Mahler introduced the concept of perfect systems in the theory of
simultaneous Hermite-Pad\'{e} approximation of analytic functions. Recently, we
proved that Nikishin systems, generated by measures with bounded support and
non-intersecting consecutive supports contained on the real line, are perfect.
Here, we prove that they are also perfect when the supports of the generating
measures are unbounded or touch at one point. As an application, we give a
version of Stieltjes' theorem in the context of simultaneous Hermite-Pad\'e
approximation.
