Subspaces of $L_p$ that embed into $L_p(\mu)$ with $\mu$ finite

TL;DR

This paper characterizes subspaces of $L_p$ spaces for $1< p < 2$ that can embed into finite measure $L_p( u)$ spaces, showing that excluding $ell_p(eth_1)$ embeddings implies embeddability into finite measure spaces.

Contribution

It proves that subspaces of $L_p$ not containing $ell_p(eth_1)$ embed into some $L_p( u)$ with finite measure, extending previous non-embedding results.

Findings

Subspaces of $L_p$ not containing $ell_p(eth_1)$ embed into finite measure $L_p$ spaces.

Generalizes non-embedding results of $ell_p(eth_1)$ to broader subspace classes.

Provides a characterization of subspace embeddability into finite measure $L_p$ spaces.

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 $\mu$.