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

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
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