Remarks on Talagrand’s deviation inequality for Rademacher functions

where σ is the Lipschitz constant of the extension of f and P is the natural probability on {0, 1}. Here we extend this inequality to more general product probability spaces; in particular, we prove the same inequality for {0, 1} with the product measure ((1− η)δ0 + ηδ1) . We believe this should be useful in proofs involving random selections. As an illustration of possible applications we give a simple proof (though not with the right dependence on ε) of the Bourgain, Lindenstrauss, Milman result [BLM] that for 1 ≤ r < s ≤ 2 and ε > 0, every n-dimensional subspace of Ls (1+ ε)embeds into lr with N = c(r, s, ε)n. The main results. For i = 1, . . . , n let (Xi, ‖ · ‖i), be normed spaces, let Ωi be a finite subset of Xi with diameter at most one and let Pi be a probability measure on Ωi. Define