On constructions of isometric embeddings of nonseparable \(L^p\) spaces, \(0 \lt p \leq 2\)

Let \(J\) be an infinite set. Let \(X\) be a real or complex \(\sigma\)-order continuous rearrangement invariant quasi-Banach function space over \((\{0, 1\}^J,\ \mathcal{B}^J,\ \lambda_J)\), the product of \(J\) copies of the measure space \((\{0, 1\},\ 2^{0,1},\ \frac{1}{2} \delta_0 + \frac{1}{2}\delta_1)\). We show that if \(0 \lt p \lt 2\) and \(X\) contains a function \(f\) with the decreasing rearrangement \(f^∗\) such that \(f^∗(t) \gt t^{-\frac{1}{p}}\) for every \(t\in (0, 1)\), then it contains an isometric copy of the Lebesgue space \(L^p (\lambda_J)\). Moreover, if \(X\) contains a function \(f\) such that \(f^∗(t) \gt \sqrt{|\text{ln}(t)|}\) for every \(t\in (0, 1)\), then it contains an isometric copy of the Lebesgue space \(L^2(\lambda_J)\).