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.