  We address the classical and quantum marginal problems, namely the question
of simultaneous realizability through a common probability density in phase
space of a given set of compatible probability distributions. We consider only
distributions authorized by quantum mechanics, i.e. those corresponding to
complete commuting sets of observables. For four-dimensional phase space with
position variables qi and momentum variables pj, we establish the two following
points: i) given four compatible probabilities for (q1,q2), (q1,p2), (p1,q2)
and (p1,p2), there does not always exist a positive phase space density
rho({qi},{pj}) reproducing them as marginals; this settles a long standing
conjecture; it is achieved by first deriving Bell-like inequalities in phase
space which have their own theoretical and experimental interest. ii) given
instead at most three compatible probabilities, there always exist an
associated phase space density rho({qi},{pj}); the solution is not unique and
its general form is worked out. These two points constitute our ``three
marginal theorem''.
