Pointwise versus equal (quasi-normal) convergence via ideals (Q465194)

Let \(\mathcal{I},\mathcal{J}\) be proper ideals on \(\mathbb{N}\). Let \(f_n\) \((n\in \mathbb{N})\) and \(f\) be real-valued functions defined on a set \(X\).NEWLINENEWLINEThe sequence \((f_n)\) is \(\mathcal{I}\)-pointwise convergent to \(f\) if \(\{n: |f_n(x)-f(x)|\geq \varepsilon\}\in \mathcal{I}\) for every \(x\in X\) and \(\varepsilon >0\).NEWLINENEWLINEThe sequence \((f_n)\) is \((\mathcal{I},\mathcal{J})\)-equally convergent to \(f\) if there exists a sequence of positive reals \((\varepsilon_n)\) which is \(\mathcal{J}\)-pointwise convergent to \(0\) and \(\{n: |f_n(x)-f(x)|\geq \varepsilon_n\}\in \mathcal{I}\) for every \(x\in X\).NEWLINENEWLINEIn this paper, the authors prove the following theorem:NEWLINENEWLINE{Theorem.} Let \(\mathcal{I},\mathcal{J}\) be ideals on \(\mathbb{N}\). For every set \(X\) the following assertions are equivalent.NEWLINENEWLINE(1) For every sequence \((f_n)\) of real-valued functions defined on \(X\), if \((f_n)\) is \(\mathcal{I}\)-pointwise convergent to \(f\), then \((f_n)\) is \((\mathcal{I},\mathcal{J})\)-equally convergent to \(f\).NEWLINENEWLINE(2) For every family \(\{E_{n}^{\alpha}:n\in \mathbb{N}, \alpha<|X|\}\subseteq \mathcal{I}\) such that \(E_{n}^{\alpha}\cap E_{k}^{\alpha}=\emptyset\) for \(n\neq k\), \(\alpha<|X|\), there exists a partition \(\{A_n: n\in \mathbb{N}\}\subseteq \mathcal{J}\) of \(\mathbb{N}\) such that NEWLINE\[NEWLINE \bigcup_{n\in \mathbb{N}}\left(A_n\cap \bigcup_{i\leq n}E_{i}^{\alpha}\right)\in \mathcal{I} NEWLINE\]NEWLINE for every \(\alpha<|X|\).