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  • Author: Władysław Wilczyński x
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W. Orlicz in 1951 has observed that if {fn(·, y)}n N converges in measure to f(·, y) for each y ∈ [0, 1], then {fn}n N converges in measure to f on [0, 1] × [0, 1]. The situation is different for the convergence in category even if we assume the convergence in category of sequences {fn(·, y)}n N for each y ∈ [0, 1] and {fn(x, ·)}n N for each x ∈ [0, 1].


The classical Lebesgue density theorem says that almost each point of a measurable set A is a density point of A. It is well known that the density point of a measurable set A can be described in terms of the convergence in measure of a sequence of characteristic functions of sets similar to A. In this note it is shown that in the Lebesgue density theorem the convergence in measure cannot be replaced by the convergence almost everywhere.


The paper deals with two ideals of subsets of defined with the help of density points and its category analogue. We present basic properties of these ideals and show that they are incomparable under inclusion.


Ψ-density point of a Lebesgue measurable set was introduced by Taylor in [Taylor S.J., On strengthening the Lebesgue Density Theorem, Fund. Math., 1958, 46, 305–315] and [Taylor S.J., An alternative form of Egoroff’s theorem, Fund. Math., 1960, 48, 169–174] as an answer to a problem posed by Ulam. We present a category analogue of the notion and of the Ψ-density topology. We define a category analogue of the Ψ-density point of the set A at a point x as the Ψ-density point at x of the regular open representation of A.


In this note, we introduce the notion of regular density. Next, we prove that x ℝ is the regular density point of a measurable set A if and only if it is an O’Malley point of A.


We study the properties of category density topology of the plane generated by a restricted convergence in the category of double sequences of characteristic functions

and more interesting topology generated by a strict convergence in the category of the same sequences, which is a natural modification of a previous one. Similar problems for measure density were considered in [M. Filipczak, W. Wilczy´nski: Strict density topology on the plane. Measure case (in preparation)].


Let C 0 denote the set of all non-decreasing continuous functions f : (0, 1] →(0, 1] such that limx →0+ ƒ(x) = 0 and ƒ(x) ≤ x for x ∈(0, 1] and let A be a measurable subset of the plane. We define the notion of a density point of A with respect to ƒ. This is a starting point to introduce the mapping D ƒ defined on the family of all measurable subsets of the plane, which is so-called lower density. The mapping D ƒ leads to the topology T ƒ, analogously as for the density topology. The properties of the topologies T ƒ are considered.


In this paper we shall study a density-type topology generated by the convergence everywhere except for a finite set similarly as the classical density topology is generated by the convergence in measure. Among others it is shown that the set of finite density points of a measurable set need not be measurable.