  In the approximate integration some inequalities between the quadratures and
the integrals approximated by them are called \emph{extremalities}. On the
other hand, the set of all quadratures is convex. We are trying to find
possible connections between extremalities and extremal quadratures (in the
sense of extreme points of a~convex set). Of course, the quadratures are the
integrals \wrt~discrete measures and, moreover, a~quadrature is extremal if and
only if the associated measure is extremal. Hence the natural problem arises to
give some description of extremal measures with prescribed moments in the
general (not only discrete) case. In this paper we deal with symmetric measures
with prescribed first four moments. The full description (with no symmetry
assumptions, and/or not only four moments are prescribed and so on) is far to
be done.
