  By generalizing the Fujikawa approach, we show in the path-integral
formalism: (1) how the infinitesimal variation of the fermion measure can be
integrated to obtain the full anomalous chiral action; (2) how the action
derived in this way can be identified as the Chern-Simons term in five
dimensions, if the anomaly is consistent; (3) how the regularization can be
carried out, so as to lead to the consistent anomaly and not to the covariant
anomaly. Our method uses Schwinger's ``proper-time'' representation of the
Green's function and the gauge invariant point-splitting technique. We find
that the consistency requirement and the point-splitting technique allow both
an anomalous and a non-anomalous action. In the end, the nature of the vacuum
determines whether we have an anomalous theory, or, a non-anomalous theory
