On the discrete ideal convergence of sequences of quasi-continuous functions

Ideal equal Baire classes

“…Recently, Natkaniec and Szuca (see [18] and [19]) obtained similar results in the case of quasi-continuous functions instead of continuous functions. Namely, they characterized Baire systems generated by the family of quasi-continuous functions in the case of ideal convergence and ideal discrete convergence for all Borel ideals and metric Baire spaces.…”

“…Proof. By [19,Theorem 9], every f ∈ P W D 0 (X) is an I-discrete limit of a sequence of quasi-continuous functions. Then, by Lemma 3.4, f ∈ (I, J ) (QC (X)).…”

“…All earlier results from this area (cf. [5], [8], [16], [18] and [19]) have a similar structure: they state that for any α < ω 1 and a Borel ideal the Baire class α (generated by continuous or quasicontinuous functions) with respect to some notion of ideal convergence is equal to one of the Baire classes (generated by the same family of functions) with respect to classical (i.e., not involving ideals) counterpart of the same notion of convergence. We show that the Baire system (generated by continuous or quasi-continuous functions) with respect to ideal equal convergence can be equal to the Baire system (generated by the same family of functions) but with respect to classical convergence (not classical equal convergence).…”

“…Since the ideals I ↾ A and P(A) are orthogonal, we have Baire (X) ⊆ R X ⊆ (I ↾ A, P(A)) (QC (X)) by Lemma 3.8.Therefore, it suffices to show that Baire (X) ⊆ (I ↾ (ω \ A), J ↾ (ω \ A)) (QC (X)).Let f : X → R be a function possessing the Baire property. By Theorem 4.4 and[19, Proposition 16], there is a sequence (g (n,m) ) (n,m)∈ω 2 of quasi-continuous functions WR-discretely convergent to f . By Theorem 4.4, there also is a bijection π : ω \ A → ω 2 with π −1 [M ] ∈ I ↾ (ω \ A) for each M ∈ WR.…”