Continuous logic and embeddings of Lebesgue spaces

We use the compactness theorem of continuous logic to give a new proof that $L^r([0,1];\mathbb{R})$ isometrically embeds into $L^p([0,1];\mathbb{R})$ whenever $1\le p\le r\le 2$. We will also give a proof for the complex case. This will involve a new characterization of complex $L^p$ spaces based on Banach lattices.