Isometric classification of the $L^p$-spaces of infinite dimensional Lebesgue measure

Abstract

We investigate the isometric structure of $L^p$-spaces for the infinite-dimensional Lebesgue measure $({\mathbb{R}}^{\mathbb{N}},\mu )$. Under the continuum hypothesis (CH) we prove $L^p(\mu )\cong {\ell }^p(\mathfrak{c},L^p[0,1])$, where $\mathfrak{c}$ denotes the cardinality of the continuum, and without CH we obtain an isometric, complemented copy of ${\ell }^p(\mathfrak{c},L^p[0,1])$ inside $L^p(\mu )$. In a general framework, we characterize precisely when $L^p(\nu )\cong {\ell }^p(\kappa ,L^p[0,1])$ and classify all such isometries.