  We consider the construction and classification of some new mathematical
objects, called ergodic spatial stationary processes, on locally compact
Abelian groups, which provide a natural and very general setting for studying
diffraction and the famous inverse problems associated with it. In particular
we can construct complete families of solutions to the inverse problem from any
given pure point measure that is chosen to be the diffraction. In this case
these processes can be classified by the dual of the group of relators based on
the set of Bragg peaks, and this gives a solution to the homometry problem for
pure point diffraction.
  An ergodic spatial stationary process consists of a measure theoretical
dynamical system and a mapping linking it with the ambient space in which
diffracting density is supposed to exist. After introducing these processes we
study their general properties and link pure point diffraction to almost
periodicity.
  Given a pure point measure we show how to construct from it and a given set
of phases a corresponding ergodic spatial stationary process. In fact we do
this in two separate ways, each of which sheds its own light on the nature of
the problem. The first construction can be seen as an elaboration of the
Halmos--von Neumann theorem, lifted from the domain of dynamical systems to
that of stationary processes. The second is a Gelfand construction obtained by
defining a suitable Banach algebra out of the putative eigenfunctions of the
desired dynamics.
