Władysław Wilczyński

Simple density topology

We introduce a new topology on the real line generated by the simple density points for measure. We show also that a simple category density point does not lead to a new notion.

Cauchy condition for the convergence in category

It is well known that the sequence {fn}n∈N of real measurable functions converges in measure to some measurable function f if and only if {fn}n∈N is fundamental in measure. In this note we introduce the notion of sequence fundamental in category in this manner such that the sequence {fn}n∈N of real functions having the Baire property converges in category to some function f having the Baire property if and only if {fn}n∈N is fundamental in category.

Spaces of measurable functions

The work deals with the spaceM ofS-measurable real functions equipped with the convergence structure introduced by the first author (so called convergence with respect to the σ-idealI⊂S). Among others there is proved thatM is a topological (metric) space if and only ifS/I is a topological (metric) space with respect to the order convergence.

On points of the regular density

ABSTRACT 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.

Strict density topology of the plane. Category case

Abstract 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)].

Density topologies on the plane between ordinary and strong

Abstract Let C0 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 Dƒ defined on the family of all measurable subsets of the plane, which is so-called lower density. The mapping Dƒ leads to the topology Tƒ, analogously as for the density topology. The properties of the topologies Tƒ are considered.

Absolute continuity of functions of two variables and the theorem of Burenkoff

SummaryFor real functions of real variable the following theorem is true: If Φ is an absolutely continuous function andf has an absolutely continuous derivativef′, then Φ(f(x))·f′(x) is absolutely continuous. In this note we shall show by simple examples that analogous theorem for absolutely continuous function Φ of two variables and for differentiable plane transformationf having an absolutely continuous JacobianJ does not hold. We shall consider absolute continuity in the sense of Tonelli and of the rectangle function.RiassuntoSi prova che per funzioni reali di due variabili reali il teorema di W. U. Burenkoff [1] non sussiste ove si assuma quale definizione di funzione assolutamente continua quella di Tonelli o quella di Hardy.