Nikishin systems are perfect. The case of unbounded and touching supports

K. Mahler introduced the concept of perfect systems in the theory of simultaneous Hermite–Padé 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 the Stieltjes theorem in the context of simultaneous Hermite–Padé approximation.