  The result of performing integrations over connection type variables in the
path integral for the discrete field theory may be poorly defined in the case
of non-compact gauge group with the Haar measure exponentially growing in some
directions. This point is studied in the case of the discrete form of the first
order formulation of the Einstein gravity theory. Here the result of interest
can be defined as generalized function (of the rest of variables of the type of
tetrad or elementary areas) i. e. a functional on a set of probe functions. To
define this functional, we calculate its values on the products of components
of the area tensors, the so-called moments. The resulting distribution (in
fact, probability distribution) has singular ($\delta$-function-like) part with
support in the nonphysical region of the complex plane of area tensors and
regular part (usual function) which decays exponentially at large areas. As we
discuss, this also provides suppression of large edge lengths which is
important for internal consistency, if one asks whether gravity on short
distances can be discrete. Some another features of the obtained probability
distribution including occurrence of the local maxima at a number of the
approximately equidistant values of area are also considered.
