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 Fσ Radon–Nikodym ideal, that is, an ideal Z⊆P(N) together with a Borel Z-approximate homomorphism which is not Z-approximable by a continuous true homomorphism .