Generalized Egorov’s Statement for ideals

Generalized Egorov’s statement for ideals, Real Anal. Exchange, 42(2):269–282, 2017. (arXiv)

Most mathematicians know the classic Egorov Theorem. It states that given a sequence of measurable functions (let restrict our attention to real functions) on a measure space, which is pointwise convergent on a measurable set A and \varepsilon>0, one can find B\subseteq A of measure \varepsilon such that the sequence converges uniformly on A\setminus B.

It is interesting whether we can drop the assumption about measurability of the functions. A statement which says that the theorem holds without this assumption for any sequence of functions f_{n}\colon [0,1]\to\mathbb{R}, is called the generalized Egorov’s statement. T. Weiss has proven that it is independent from ZFC and then R. Pinciroli studied this independence more systematically. For example non(N)<\mathfrak{b} implies that the statement holds, but if non(N)=\mathfrak{b}, it fails.

But we can also define a version of convergence of a sequence of functions with respect to a given ideal on \omega. There are different types of convergence here and pointwise and uniform convergence are the most studied. We can ask whether the classic Egorov’s Theorem (with the measurability assumption) holds for given ideal. The answer is usually negative in the case of uniform and pointwise convergence for analytic P-ideals. But one can also define other types of convergence, e.g. equi-ideal convergence. In the case of analytic P-ideal so called week Egorov’s Theorem for ideals (between pointwise and equi-ideal convergence) has been proven by N. Mrożek. And there are more examples of positive results.

Therefore we have asked whether in the case of convergence with respect to an ideal we can drop the assumption of measurability. The work is still in progress (for different types of convergence), but I have already proven that the methods proposed by R. Pincoroli in the simple convergence case can be successfully adjusted to the case of weak Egorov’s Theorem for analytic P-ideals.