The $\mathcal{I}$-almost constant convergence of sequences of real functions.

Abstract

Let \(T\) be an infinite set and \(\mathcal{I}\) be a fixed ideal on \(T\). We introduce and study the notion of almost constant convergence of sequences \(\{ f_t\colon t\in T\}\) of real functions with respect to the ideal \(\mathcal{I}\). This notion generalizes discrete convergence, transfinite convergence, and \(\omega_2\)-convergence. In particular, we consider the question when a given family of functions (e.g., continuous, Baire class 1, Borel measurable, Lebesgue measurable, or functions with the Baire property) is closed with respect to this kind of convergence.