Reasonable non-Radon–Nikodym ideals

Following a research line suggested by Ilijas Farah, we prove that for any abelian Polish σ-compact group H there exists an F σ Radon–Nikodym ideal, that is, an ideal Z ⊆ P ( N ) together with a Borel Z -approximate homomorphism f : H → H N which is not Z -approximable by a continuous true homomorphism g : H → H N .