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.
