Embedding measure spaces

‎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-Cech compactification of a completely regular topological space‎. ‎Under certain conditions the construction simplifies‎. ‎Examples are given when this simplification occurs‎.