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

TL;DR

This work analyzes the isometric structure of $L^{p}$-spaces associated with the infinite-dimensional Lebesgue measure $\mu$ on $\mathbb{R}^{\mathbb{N}}$, a non-$\sigma$-finite and not locally finite measure. Under the Continuum Hypothesis, it shows the precise isometric classification $L^{p}(\mu)\cong \ell^{p}(\mathfrak{c},L^{p}[0,1])$, while in ZFC alone it still yields an isometric complemented copy of this space inside $L^{p}(\mu)$ via a cube decomposition. The authors develop a general framework by decomposing $L^{p}(\mu)$ into two complemented parts, $G^{p}$ and $B^{p}$, and they prove both parts contain isometric copies of $\ell^{p}(\mathfrak{c},L^{p}[0,1])$, establishing a robust structural picture even without CH. They further characterize when $L^{p}(\nu)\cong \ell^{p}(\kappa,L^{p}[0,1])$ for relatively nonatomic $\nu$, describe the isometric forms of such isomorphisms, and discuss a conjecture that the central CH-based isomorphism may be independent of ZFC.

Abstract

We investigate the isometric structure of $L^{p}$-spaces for the infinite-dimensional Lebesgue measure $(\mathbb{R}^{\mathbb{N}},μ)$. Under the continuum hypothesis (CH) we prove $L^{p}(μ)\cong \ell^{p}(\mathfrak{c},L^{p}[0,1])$, 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 $L^{p}(μ)$. In a general framework, we characterize precisely when $L^{p}(ν)\cong \ell^{p}(κ,L^{p}[0,1])$ and classify all such isometries.

Theorems & Definitions (51)

• Theorem 1.1
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Lemma 2.3
• proof
• Theorem 2.4
• proof
• Theorem 2.5