A generalized Egorov’s statement for ideals. (English) Zbl 1392.28006

Korch, Mikhail

Real Anal. Exch. 42, No. 2, 269-282 (2017).

The author presents various versions of the classic Egorov’s theorem. One of these says: Let \(I\) be a countable generated ideal on \(\omega\) and \(f_n:[0,1]\rightarrow [0,1]\), \(n\in\omega\), be a sequence of Lebesgue-measurable functions such that \(f_n\rightarrow_I 0\) (i.e., \(\{n\in\omega: f_n(x)\geq\epsilon\}\in I\) for any \(\epsilon >0\) and \(x\in [0,1]\)). Then there exists for any \(\delta>0\) a set \(B\subseteq [0,1]\) of Lebesgue measure \(\leq \delta\) such that \(f_n\rightrightarrows_I 0\) on \([0,1]\setminus B\) (i.e., for any \(\epsilon>0\) there is a set \(A\in I\) such that \(\{n\in\omega: f_n(x)\geq\epsilon\}\subseteq A\) for any \(x\in [0,1]\setminus B\)). An analogous theorem holds replacing the assumption that the function \(f_n\) are measurable by the assumption that the lowest cardinality of a Lebesgue non-null set is smaller than the lowest cardinality of a family of sequences of natural numbers unbounded in the sense of the order of eventual domination.

Reviewer: Hans Weber (Udine)

MSC:

28A20	Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
26A03	Foundations: limits and generalizations, elementary topology of the line
03E20	Other classical set theory (including functions, relations, and set algebra)
03E35	Consistency and independence results
40A35	Ideal and statistical convergence

Keywords:

Egorov’s theorem; generalized Egorov’s statement; ideal convergence