On the independence of a generalized statement of Egoroff's theorem from ZFC after T. Weiss (Q2371939)

The paper deals with the following statement GES generalizing Egoroff's well-known theorem: For every sequence \(\{f_n:n\in\mathbb N\}\) of real functions converging pointwise to zero and for every \(\eta>0\) there is a~set \(A\subseteq[0,1]\) with outer measure \(\mu^*(A)>1-\eta\) such that \(\{f_n\}\) converges uniformly on~\(A\). T.~Weiss has proved the independence of this statement of ZFC. In the paper under review the author proves that if \(\text{non}(\mathcal N)<\mathfrak b\), then GES holds. On the other hand GES fails if any of the following hypotheses holds: (1)~\(\text{non}(\mathcal N)=\mathfrak d=\mathfrak c\); (2)~there exists a~\(\mathfrak c\)-Luzin set and \(\text{non}(\mathcal N)=\mathfrak c\); (3)~there exists a~\(\mathfrak c\)-Luzin set and \(\mathfrak c\)~is a~regular cardinal.