  Averaging physical quantities over Lie groups appears in many contexts across
the rapidly developing branches of physics like quantum information science or
quantum optics. Such an averaging process can be always represented as
averaging with respect to a finite number of elements of the group, called a
finite averaging set. In the previous research such sets, known as $t$-designs,
were constructed only for the case of averaging over unitary groups (hence the
name unitary $t$-designs). In this work we investigate the problem of
constructing finite averaging sets for averaging over general non-compact
matrix Lie groups, which is much more subtle task due to the fact that the the
uniform invariant measure on the group manifold (the Haar measure) is infinite.
We provide a general construction of such sets based on the Cartan
decomposition of the group, which splits the group into its compact and
non-compact components. The averaging over the compact part can be done in a
uniform way, whereas the averaging over the non-compact one has to be endowed
with a suppresing weight function, and can be approached using generalised
Gauss quadratures. This leads us to the general form of finite averaging sets
for semisimple matrix Lie groups in the product form of finite averaging sets
with respect to the compact and non-compact parts. We provide an explicit
calculation of such sets for the group $SL(2, \mathbb{C})$, although our
construction can be applied to other cases. Possible applications of our
results cover finding finite ensambles of random operations in quantum
information science and quantum optics, which can be used in constructions of
randomised quantum algorithms, including optical interferometric
implementations.
