  We introduce a symmetrization technique that allows us to translate a problem
of controlling the deviation of some functionals on a product space from their
mean into a problem of controlling the deviation between two independent copies
of the functional. As an application we give a new easy proof of Talagrand's
concentration inequality for empirical processes, where besides symmetrization
we use only Talagrand's concentration inequality on the discrete cube
{-1,+1}^n. As another application of this technique we prove new
Vapnik-Chervonenkis type inequalities. For example, for VC-classes of functions
we prove a classical inequality of Vapnik and Chervonenkis only with
normalization by the sum of variance and sample variance.

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