Nonmeasurability in Banach spaces

We show that for a $\sigma $-ideal $\ci$ with a Steinhaus property defined on Banach space, if two non-homeomorphic Banach with the same cardinality of the Hamel basis then there is a $\ci$ nonmeasurable subset as image by any isomorphism between of them. Our results generalize results from [2]

💡 Research Summary

The paper investigates the phenomenon of non‑measurability in Banach spaces when the underlying σ‑ideal possesses the Steinhaus property. The authors consider two Banach spaces (X) and (Y) that are not homeomorphic but share the same cardinality of a Hamel basis. Under these hypotheses they prove that any linear isomorphism (T\colon X\to Y) necessarily maps some subset of (X) that is not (\mathcal I)-measurable onto a set that is also not (\mathcal I)-measurable in (Y). This result extends earlier work (reference