On the \(\omega_1\) limits of subsets of the real line (Q1046905)

The set \(A=\bigcup_{\alpha<\omega_1}\bigcap_{\beta<\alpha}A_\beta\) is said to be the \(\omega_1\)-limit of the \(\omega_1\)-sequence of sets \(\langle A_\alpha:\alpha<\omega_1\rangle\). The authors point out that in the known Bell-Kunen model every set of reals is an \(\omega_1\)-limit of \(G_\delta\) sets. On the other hand they prove that assuming \(\text{cov}(\mathcal M)>\omega_1\) (i.e., the real line cannot be covered by \(\omega_1\) meager sets) there is a~set of reals which is not an \(\omega_1\)-limit of measurable sets. This solves two problems of \textit{T. Natkaniec} and \textit{J. Wesołowska} [Acta Math. Hung. 90, No. 4, 333--350 (2001; Zbl 0980.26001)]