Preregularization and the path integral approach to the chiral anomaly

We explore the connection between perturbative and non-perturbative (path-integral) approaches to the axial anomaly. In particular, we show how the Jacobian associated with the fermionic measure corresponding to local axial transformations may be calculated directly from shift-of-integration-variable surface terms in four Euclidean dimensions. No regularization (explicit parametrization of UV infinities) is required in this approach, but invariance of the Jacobian under vector gauge transformations (i.e. preregularization) is required to remove a variable-of-integration ambiguity within the expression for the Jacobian of the fermionic measure.