  For a given measure space $(X,{\mathscr B},\mu)$ we construct all measure
spaces $(Y,{\mathscr C},\lambda)$ in which $(X,{\mathscr B},\mu)$ is
embeddable. The construction is modeled on the ultrafilter construction of the
Stone--\v{C}ech compactification of a completely regular topological space.
Under certain conditions the construction simplifies. Examples are given when
this simplification occurs.
