The isomorphism theorem for linear fragments of continuous logic

The ultraproduct construction is generalized to $p$-ultramean constructions ($1\leqslant p<\infty$) by replacing ultrafilters with finitely additive measures. These correspond the linear fragments $\mathscr L^p$ of continuous logic. A powermean variant Keisler-Shelah isomorphism theorem proved for L^p$. It then that L^p$-sentences (and their approximations) are exactly those sentences logic which preserved such constructions. Some other applications also given.