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