Corrigendum in: a generalization of density topology and on generalization of the density topology on the real line (Q2392536)

Let \(\mathcal A_d\) be a family of measurable subsets of \([-1,1]\) that have density one at the point \(0\) (here \(d\) means the Lebesgue real measure). In this short note, the author provides a new definition of \(\mathcal A_d\)-density point of a measurable set \(A\subset \mathbb R\), inspired by the notion of segment density point introduced in his paper [the author, in: Real functions, density topology and related topics. Dedicated to Professor Władysław Wilczyński on the occasion of his 65th birthday. Łódź: Wydawnictwo Uniwersytetu Łódzkiego. 29--36 (2011; Zbl 1236.54004)]. With this new definition, all the results presented in his papers [Real Anal. Exch. 32, 349--358 (2007; Zbl 1135.28001)] and [Real Anal. Exch. 33, 199--214 (2008; Zbl 1151.54005)] stay valid and, even, some proofs of the mentioned papers can be shorter. It is worth pointing out that this new definition allows the author to fill a gap in the above mentioned [2007, loc. cit.], where it was erroneously established that a certain associate operator \(\Phi_{\mathcal A_d}\) is monotonic. In the present paper, the author gives a counterexample to this monotonic property for the ancient definition by finding two measurable sets \(A\subset E\) for which \(0\in\Phi_{\mathcal A_d}(A)\setminus \Phi_{\mathcal A_d}(E).\) Nevertheless, with the new definition now the operator is already a lower density operator.