The problem of formulating the chiral fermion field on space-time lattices is discussed from the viewpoint of functional integrals. We argue that the functional integral measure provided by any single lattice is not sufficent to yield solutions to agree with the continuum field theory. The approaches of using an ensemble of random-block lattices (RBL) as well as CFL random lattices to provide a proper functional integral measure are investigated. For a massless fermion field interacting with a background abelian gauge field in 2D, the RBL randomization gives results in very good agreement with the continuum field theory, while the CFL randomization fails to yield the correct axial-vector current and axial anomaly.
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