Abstract
We improve and generalize the result of Kirchheim and Natkaniec [Kirchheim, B., Natkaniec, T.: Exceptional directions for Sierpi´nski’s nonmeasurablesets, Fund. Math. 140 (1992), 237-245] by proving that there exists D ⊂[0, π) of full outer measure such that the class ∩θ∈DG2(θ) ∩∩θ∈[0,π)G3(θ) contains a nonmeasurable set. The present result is obtained within ZFC.