On the independence of a generalized statement of Egoroff’s theorem from ZFC after T. Weiss. (English) Zbl 1120.03030

Pinciroli, Roberto

Real Anal. Exch. 32(2006-2007), No. 1, 225-232 (2007).

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.

Reviewer: Miroslav Repický (Košice)

MSC:

03E35	Consistency and independence results
03E17	Cardinal characteristics of the continuum
28A20	Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
40A30	Convergence and divergence of series and sequences of functions

Keywords:

cardinal characteristics of the continuum; Luzin sets