We still consider the binary classification problem on X, and we denote fn: F(X,Y) a classifier. We say that it is universally consistent in probability, if V > 0, sup P(|Lp(n) Lp(fp)| > 6) na 0, (0.2) where we precise in index the probability distribution (be careful, for the convergence in probability and the computations of expected risk). We suppose that is finite: Card(X) = K. 1. What is the cardinality of F(X,Y)? 2. Recall the risk bound proved in the class for the ERM fn,ERM with respect to F(X,Y). Can we dedude that fn,ERM is universally consistent ? 3. We now suppose that K = Kn depends on the sample size. Prove that if K is sub-linear in n, then fn,ERM est universally consistent.