The isomorphism theorem for linear fragments of continuous logic

The ultraproduct construction is generalized to p ‐ultramean constructions ( 1 ⩽ p < ∞ ) by replacing ultrafilters with finitely additive measures. These constructions correspond to the linear fragments L p of continuous logic and are very close to theL p ( R )constructions in real analysis. A powermean variant of the Keisler‐Shelah isomorphism theorem is proved for L p . It is then proved that L p ‐sentences (and their approximations) are exactly those sentences of continuous logic which are preserved by such constructions. Some other applications are also given.