Nonmeasurability in Banach spaces

TL;DR

This paper investigates nonmeasurable sets in Banach spaces, demonstrating that under certain conditions, isomorphic Banach spaces with the same basis cardinality have nonmeasurable images, extending previous results.

Contribution

It generalizes earlier findings by establishing nonmeasurability of images under isomorphisms in a broader class of Banach spaces with a Steinhaus property.

Findings

Non-homeomorphic Banach spaces with same basis cardinality have nonmeasurable images.

Extension of previous results to a wider class of Banach spaces with Steinhaus property.

Identification of conditions under which nonmeasurable sets exist in Banach space isomorphisms.

Abstract

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]