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.
