  L\"uscher's recent formulation of Abelian chiral gauge theories on the
lattice, in the vacuum (or perturbative) sector in infinite volume, is
reinterpreted in terms of the lattice covariant regularization. The gauge
invariance of the effective action and the integrability of the gauge current
in anomaly-free cases become transparent. The real part of the effective action
is simply one-half that of the Dirac fermion and, when the Dirac operator
behaves properly in the continuum limit, the imaginary part in this limit
reproduces the $\eta$-invariant.
