Measurability in C(2^k) and Kunen cardinals

TL;DR

This paper explores the relationship between Kunen cardinals and measurability properties in Banach spaces of continuous functions on Cantor cubes, establishing a key equivalence involving sigma-algebras.

Contribution

It proves that a cardinal is Kunen if and only if certain sigma-algebras on the associated Banach space coincide, linking set-theoretic and functional analysis properties.

Findings

Kunen cardinals characterized by sigma-algebra equivalence.

Equivalence between Kunen cardinals and Baire/Borel sigma-algebra coincidence.

Additional links between Kunen cardinals and Banach space measurability.

Abstract

A cardinal k is called a Kunen cardinal if the sigma-algebra on k x k generated by all products AxB, coincides with the power set of k x k. For any cardinal k, let C(2^k) be the Banach space of all continuous real-valued functions on the Cantor cube 2^k. We prove that k is a Kunen cardinal if and only if the Baire sigma-algebra on C(2^k) for the pointwise convergence topology coincides with the Borel sigma-algebra on C(2^k) for the norm topology. Some other links between Kunen cardinals and measurability in Banach spaces are also given.