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Ideal limits of sequences of continuous functions

Miklós Laczkovich, Ireneusz Recław (2009)

Fundamenta Mathematicae

We prove that for every Borel ideal, the ideal limits of sequences of continuous functions on a Polish space are of Baire class one if and only if the ideal does not contain a copy of Fin × Fin. In particular, this is true for $F_{σ δ}$ ideals. In the proof we use Borel determinacy for a game introduced by C. Laflamme.

Individual ergodic theorem in a regular space

Ervín Hrachovina (1987)

Mathematica Slovaca

Infinite Iterated Function Systems: A Multivalued Approach

K. Leśniak (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

We prove that a compact family of bounded condensing multifunctions has bounded condensing set-theoretic union. Compactness is understood in the sense of the Chebyshev uniform semimetric induced by the Hausdorff distance and condensity is taken w.r.t. the Hausdorff measure of noncompactness. As a tool, we present an estimate for the measure of an infinite union. Then we apply our result to infinite iterated function systems.

Inserting measurable functions precisely

Javier Gutiérrez García, Tomasz Kubiak (2014)

Czechoslovak Mathematical Journal

A family of subsets of a set is called a $σ$ -topology if it is closed under arbitrary countable unions and arbitrary finite intersections. A $σ$ -topology is perfect if any its member (open set) is a countable union of complements of open sets. In this paper perfect $σ$ -topologies are characterized in terms of inserting lower and upper measurable functions. This improves upon and extends a similar result concerning perfect topologies. Combining this characterization with a $σ$ -topological version of Katětov-Tong...

Integral extension with locally-integral seminorm

M. Díaz Carillo, F. Muňoz Rivas (1990)

Acta Universitatis Carolinae. Mathematica et Physica