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In the gauge invariant formulation of U(1) chiral lattice gauge theoriesbased on the Ginsparg-Wilson relation, the gauge field dependence of thefermion measure is determined through the so-called measure term. We derive aclosed formula of the measure term on the finite volume lattice. The Wilsonline degrees of freedom (torons) of the link field are treated separately totake care of the global integrability. The local counter term is explicitlyconstructed with the local current associated with the cohomologically trivialpart of the gauge anomaly in a finite volume. The resulted formula is veryclose to the known expression of the measure term in the infinite volume with asingle parameter integration, and would be useful in practical implementations.

A simple construction of fermion measure term in U(1) chiral lattice gauge theories with exact gauge invariance