Subspaces of $L_p$ that embed into $L_p(μ)$ with $μ$ finite

Abstract

Enflo and Rosenthal proved that $\ell_p(\aleph_1)$, $1 < p < 2$, does not (isomorphically) embed into $L_p(μ)$ with $μ$ a finite measure. We prove that if $X$ is a subspace of an $L_p$ space, $1< p < 2$, and $\ell_p(\aleph_1)$ does not embed into $X$, then $X$ embeds into $L_p(μ)$ for some finite measure $μ$.