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On density with respect to equivalent measures

Bartosiewicz A., Filipczak M., Poreda T.

Demonstratio Mathematica

2010

Vol. 43, nr 1

21-28

In the paper there is disscussed a notion of a density point of a Borel subset of a metric space with respect to a Borel measure(mikro) . There are considered densities with respect to equivalent measures and density with respect to the limit of a sequence of equivalent measures.

On the comparison of the density type topologies generated by sequences and by functions

Filipczak M., Filipczak T.

Commentationes Mathematicae

2009

Vol. 49, [Z] 2

161-170

In the paper we investigate density type topologies generated by functions f satisfying condition [formula] which are not generated by any sequence.

On II-approximate limits and II-approximate smoothness

Zduńczyk R.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

2006

Vol. 46, [Z] 1

1-16

In this paper we present some results based on slightly modified idea of the I-density introduced by Władysław Wilczyński. Some theorems are generalized versions of results from [2] and [3]. We investigate properties of functions from R^X, where X is supplied with the I-density. We try to free our considerations from the assumption of Baire property, or measurability. In some cases this is not done yet. Star-marked statements still need that assumption, proofs presented here are done for Baire property, but it is possible to adapt them to measure. I-density itself does not require any structure of considered space but a metric vector space over R. However, in last section we confine ourselves to R, for we make use of R’s structure for simplicity. To find more about related topics see [4], [5], more bibliography one can find in [1] and [5].

On homeomorphisms of the density type topologies

Filipczak M., Hejduk J., Wilczyński W.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

2005

Vol. 45, [Z] 2

151--159

This paper is dealing of the homeomorphisms of the density type topologies introduced in: M. Filipczak, J. Hejduk, "On topologies associated with the Lebesgue measure", Tatra Mountains Math. Publ. 28 (2004), 187-197