continuous logic

We prove that continuous sentences preserved by the ultramean construction (a generalization of the ultraproduct construction) are exactly those sentences which are approximated by linear sentences. Continuous sentences preserved by linear elementary equivalence are exactly those sentences which are approximated in the Riesz space generated by linear sentences. Also, characterizations for linear $Delta_n$-sentences and positive linear theories will be given.

We use the ultramean construction to prove linear compactness theorem. We also extend the Rudin-Keisler ordering to maximal probability charges and characterize it by embeddings of power ultrameans.