  We prove that, if dichotomy occurs when the concentration-compactness
principle is used, the dichotomizing sequence can be choosen so that a
nontrivial part of it concentrates. Iterating this argument leads to a profile
decomposition for arbitrary sequences of bounded Borel measures. To illustrate
our results we give an application to the structure of bouded sequences in the
Sobolev space $ W^{1, p}(\R^N)$.
