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

Abstract

We prove that, for every $1\le p<\infty$, the $L^p$-space of Baker'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].