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 \(L_{\psi '}\).

Haydon，Levy和Raynaud在他们的书《随机范数空间》中证明，在给定集中具有任意截面\（\ psi（\ cdot，\ omega）\）的Musielak-Orlicz空间\（L_ \ psi \）的每个子格D可以表示为凸包的闭合（在产品拓扑中）具有随机截面\（\ psi'（\ cdot，\ omega）\）的Musielak-Orlicz空间\（L _ {\ psi'} \）的d。在本注释中，我们证明如果\（L _ {\ psi'} \）是Musielak–Orlicz空间，且在集合的凸包封闭中具有随机部分\（\ psi'（\ cdot，\ omega）\）d在膨胀下封闭，然后存在一个Musielak–Orlicz空间\（L_ \ psi \），其中D中有随机部分\（\ psi（\ cdot，\ omega）\），使得\（L _ {\ psi'} \）为的亚晶格\（L_ \ PSI \） 。此外，可以选择\（L_ \ psi \）具有与\（L _ {\ psi'} \）相同的密度特征。