Measures on perfect e-free PAC fields

TL;DR

This paper develops measures on definable sets in certain perfect PAC fields, demonstrating their definable amenability and extending results to perfect ω-free PAC fields using ultralimit measures.

Contribution

It introduces measures on definable sets in e-free perfect PAC fields and proves definable amenability of groups in these fields and in perfect ω-free PAC fields.

Findings

Measures constructed on definable sets in e-free perfect PAC fields

Definable amenability of groups in these fields established

Extension of results to perfect ω-free PAC fields via ultralimit measures

Abstract

We construct measures on definable sets in $e$-free perfect PAC fields, as well as on perfect PAC fields whose absolute Galois groups are free pro-$p$ of finite rank. We deduce the definable amenability of all groups definable in such fields. As a corollary, we additionally prove the definable amenability of all groups definable in perfect $\omega$-free PAC fields via ultralimit measures.