A question of Jarden and Shelah (Q2639103)

Let \(L=K(t)\) be an uncountable field of rational functions over \(K\), \[ S_ e(L)=\{{\bar\sigma} |\;{\bar\sigma}\in G(L)^ e,\;\tilde L^{<{\bar\sigma}>}\text{ is an RC-field}\}. \] The author gives a negative answer to the following question of \textit{M. Jarden} and \textit{S. Shelah} [Proc. Am. Math. Soc. 87, 223--228 (1983; Zbl 0508.12021)]: Is the set \(S_ e(L)\) measurable if \(K={\mathbb C}?\) Theorem. For any \(e\geq 1\) the set \(S_ e({\mathbb C}(t))\) is non-measurable.