Path-Integral Measure for Gauge-Invariant Fermion Theories

It is shown that the path-integral measure for gauge-invariant fermion theories is not invariant under the ${\gamma }_5$ transformation and the Jacobian gives rise to an extra phase factor corresponding to the Adler-Bell-Jackiw anomaly. The derivation of "anomalous" Ward-Takahashi identities by means of the variational derivative can thus be made consistent in the path-integral formalism.