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]; ℝ) isometrically embeds into L^p([0,1]; ℝ) whenever 1 ≤ p ≤ r ≤ 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.