Reasonable non--Radon--Nikodym ideals

TL;DR

The paper constructs a sigma-ideal and a Borel approximate homomorphism for any abelian Polish sigma-compact group that cannot be closely approximated by a true continuous homomorphism, highlighting limitations in approximation.

Contribution

It introduces a specific sigma-ideal and an approximate homomorphism demonstrating the failure of approximation by continuous homomorphisms in abelian Polish sigma-compact groups.

Findings

Existence of a sigma-ideal Z over N for any abelian Polish sigma-compact group H.

Construction of a Borel Z-approximate homomorphism f : H --> H^N.

f is not Z-approximable by any continuous true homomorphism g : H --> H^N.

Abstract

For any abelian Polish sigma-compact group H there exist a sigma-ideal Z over N and a Borel Z-approximate homomorphism f : H --> H^N which is not Z-approximable by a continuous true homomorphism g : H --> H^N.