  We define Weyl fermions on a finite lattice in such a way that in the path
integral the action is gauge invariant but the functional measure is not. Two
variants of such a formulation are tested in perturbative calculation of the
fermion determinant in chiral Schwinger model. We find that one of these
variants ensures restoring the gauge invariance of the nonanomalous part of the
determinant in the continuum limit. A `perfect' perturbative regularization of
the chiral fermions is briefly discussed.
