Construction of A Lattice on the completion space of an algebra and an   isomorphism to its Caratheodory Extension
  In this paper, we will show how the Caratheodory Extension process is intimately related to the metric completion process. In particular, it will be shown how one is able to construct a lattice on the completion and to obtain an isomorphism to its Caratheodory Extension. 
Jun Tanaka, Peter McLoughlin