A note on representations of Orlicz lattices

Abstract

In their book Randomly Normed Spaces, Haydon, Levy, and Raynaud proved that every sublattice of a Musielak-Orlicz space $L_\psi$ with random sections $\psi (\cdot ,\omega )$ in a given set D can be represented as a Musielak–Orlicz space $L_{\psi '}$ with random sections $\psi '(\cdot ,\omega )$ in the closure (in the product topology) of the convex hull of D. In this note we prove that if $L_{\psi '}$ is a Musielak–Orlicz space with random sections $\psi '(\cdot ,\omega )$ in the closure of the convex hull of a set D closed under dilations, then there exists a Musielak–Orlicz space $L_\psi$ with random sections $\psi (\cdot ,\omega )$ in D such that $L_{\psi '}$ is a sublattice of $L_\psi$. Furthermore, $L_\psi$ can be chosen to have the same density character as $L_{\psi '}$.