Note on infinite-dimensional $L^p$-spaces

Abstract

We prove that, for every $1\leq p<\infty$, the $L^{p}$-space of Baker&#39;s measure on $\mathbb{R}^{\mathbb{N}}$ is isometrically isomorphic to $\ell^p(\mathfrak{c},L^{p}[0,1])$ in ZFC. This solves in a negative manner the main problem stated in [Isometric classification of the $L^{p}$-spaces of infinite dimensional Lebesgue measure, Banach J. Math. Anal. 20 (2026), no. 1, Paper No. 7].