EUDML

Let X, Y be uncountable Polish spaces and let μ be a complete σ-finite Borel measure on X. Denote by K and L the families of all meager subsets of X and of all subsets of Y with μ measure zero, respectively. It is shown that the product of the ideals K and L restricted to C-sets of Selivanovskiĭ is σ-saturated, which extends Gavalec's results.

Generalized projections of Borel and analytic sets

Marek Balcerzak — 1996

Colloquium Mathematicae

For a σ-ideal I of sets in a Polish space X and for A ⊆ $X^{2}$ , we consider the generalized projection (A) of A given by (A) = x ∈ X: Ax ∉ I, where $A_{x}$ =y ∈ X: 〈x,y〉∈ A. We study the behaviour of with respect to Borel and analytic sets in the case when I is a $\sum_{2}^{0}$ -supported σ-ideal. In particular, we give an alternative proof of the recent result of Kechris showing that [ $\sum_{1}^{1} (X^{2})] = \sum_{1}^{1} (X)$ for a wide class of $\sum_{2}^{0}$ -supported σ-ideals.

A generalization of the theorem of Mauldin

Marek Balcerzak — 1985

Commentationes Mathematicae Universitatis Carolinae

On isomorphisms between $σ$ -ideals on $ω_{1}$

Marek Balcerzak — 1990

Commentationes Mathematicae Universitatis Carolinae

Classification of $s i g m a$ -ideals

Marek Balcerzak — 1987

Mathematica Slovaca

On the generalized property $(K)$

Marek Balcerzak — 1986

Mathematica Slovaca

On some ideals and related algebras of sets in the plane

Marek Balcerzak — 1989

Acta Universitatis Carolinae. Mathematica et Physica

Some examples of true $F_{σ δ}$ sets

Marek Balcerzak; Udayan Darji — 2000

Colloquium Mathematicae

Let K(X) be the hyperspace of a compact metric space endowed with the Hausdorff metric. We give a general theorem showing that certain subsets of K(X) are true $F_{σ δ}$ sets.

On multiplication in spaces of continuous functions

Marek Balcerzak; Aleksander Maliszewski — 2011

Colloquium Mathematicae

We introduce and examine the notion of dense weak openness. In particular we show that multiplication in C(X) is densely weakly open whenever X is an interval in ℝ.

On pointwise limits of sequences of $I$ -continuous functions

Marek Balcerzak; Ewa Łazarow — 1986

Commentationes Mathematicae Universitatis Carolinae

Convergence theorems for the Birkhoff integral

Marek Balcerzak; Monika Potyrała — 2008

Czechoslovak Mathematical Journal

We give sufficient conditions for the interchange of the operations of limit and the Birkhoff integral for a sequence $(f_{n})$ of functions from a measure space to a Banach space. In one result the equi-integrability of $f_{n}$ ’s is involved and we assume $f_{n} \to f$ almost everywhere. The other result resembles the Lebesgue dominated convergence theorem where the almost uniform convergence of $(f_{n})$ to $f$ is assumed.