  We consider positively supported Borel measures for which all moments exist.
On the set of compactly supported measures in this class a partial order is
defined via eventual dominance of the moment sequences. Special classes are
identified on which the order is total, but it is shown that already for the
set of distributions with compactly supported smooth densities the order is not
total. In particular we construct a pair of measures with smooth density for
which infinitely many moments agree and another one for which the moments
alternate infinitely often. This disproves some recently published claims to
the contrary. Some consequences for games with distributional payoffs are
discussed.
