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 $\{0,1\}^n.$ 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.