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

Abstract: In this paper, we investigate the isometric structure of the $L^p$-spaces associated with an infinite-dimensional analogue of the Lebesgue measure $(\mathbb{R}^{\mathbb{N}}, \mu)$. Our primary result establishes that under the Continuum Hypothesis (CH), $L^p(\mu)$ is isometrically isomorphic to $\ell^p(\mathfrak{c}, L^p[0,1])$, where $\mathfrak{c}$ denotes the cardinality of the continuum. In the general case, without assuming CH, we prove that $L^p(\mu)$ contains an isometric and complemented copy of $\ell^p(\mathfrak{c}, L^p[0,1])$. Furthermore, we characterize isometries between $L^p$-spaces and establish necessary and sufficient conditions for an isometric isomorphism of the form $L^p(\nu) \cong \ell^p(\kappa, L^p[0,1])$ for some cardinal $\kappa$. In particular, we classify all possible isometric isomorphisms between such spaces.