This is an announcement for the paper "Subspaces of $L_p$ that embed into $L_p(\mu)$ with $\mu$ finite" by William B. Johnson and Gideon Schechtman. Abstract: Enflo and Rosenthal proved that $\ell_p(\aleph_1)$, $1 < p < 2$, does not (isomorphically) embed into $L_p(\mu)$ with $\mu$ 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 $L_p(\mu)$ for some finite measure $\mu$. Archive classification: math.FA Mathematics Subject Classification: 46E30, 46B26, 46B03 Submitted from: gideon@weizmann.ac.il The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1301.4086 or http://arXiv.org/abs/1301.4086