Nonmeasurable sets with regular sections

Abstract

We improve and generalize the result of Kirchheim and Natkaniec \cite{kirchheim} by proving that there exists $D \subset [0,\pi)$ of full outer measure such that the class $\bigcap_{\theta \in D}\mathcal{G}_2(\theta) \cap \bigcap_{\theta \in [0,\pi)}\mathcal{G}_3(\theta)$ contains a nonmeasurable set. The present result is obtained within ZFC.