  If A is a finite alphabet, Z^D is a D-dimensional lattice, U is a subset of
Z^D, and mu_U is a probability measure on A^U that ``looks like'' the marginal
projection of a stationary random field on A^(Z^D), then can we ``extend'' mu_U
to such a field? Under what conditions can we make this extension ergodic,
(quasi)periodic, or (weakly) mixing? After surveying classical work on this
problem when D = 1, we provide some sufficient conditions and some necessary
conditions for mu_U to be extendible for D > 1, and show that, in general, the
problem is not formally decidable.
