Measures on perfect e-free PAC fields

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 $ω$-free PAC fields via ultralimit measures.

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