  Defining chiral lattice gauge theories in the Ginsparg-Wilson formalism is
complicated by the so-called fermion measure problem. It has been proven for
the abelian theories that smooth well-behaved fermion measure exists if and
only if the anomaly-free condition is granted, and the same was shown to hold
in perturbative theories for non-abelian gauge groups, but the non-perturbative
proof is absent. In this paper, we consider a simpler problem in 2-d and
present a proof for the existence of smooth and gauge invariant fermion measure
on the gauge field configuration space with zero field strengths for arbitrary
compact Lie groups, provided the anomaly-free conditions are satisfied. It is
conjectured that such consideration is sufficient for the unknown full proof.
