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The Hajek-Feldman dichotomy establishes that two Gaussian measures are either mutually absolutely continuous with respect to each other (and hence there is a Radon-Nikodym density for each measure with respect to the other one) or mutually singular. Unlike the case of finite-dimensional Gaussian measures, there are nontrivial examples of both situations when dealing with Gaussian stochastic processes. This article provides: (a) Explicit expressions for the optimal (Bayes) rule and the minimal classification error probability in several relevant problems of supervised binary classification of mutually absolutely continuous Gaussian processes. The approach relies on some classical results in the theory of reproducing kernel Hilbert spaces (RKHS). (b) An interpretation, in terms of mutual singularity, for the so-called near perfect classification phenomenon. We show that the asymptotically optimal rule proposed by these authors can be identified with the sequence of optimal rules for an approximating sequence of classification problems in the absolutely continuous case. (c) As an application, we discuss a natural variable selection method, which essentially consists of taking the original functional data X(t), t [0, 1] to a d-dimensional marginal (X(t(1)), ..., X(t(d))), which is chosen to minimize the classification error of the corresponding Fisher's linear rule. We give precise conditions under which this discrimination method achieves the minimal classification error of the original functional problem. Supplementary materials for this article are available online.