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Games characterizing certain families of functions

Marek Balcerzak, Tomasz Natkaniec, Piotr Szuca

Abstract

We obtain several game characterizations of Baire 1 functions between Polish spaces X, Y which extends the recent result of V. Kiss. Then we propose similar characterizations for equi-Bare 1 families of functions. Also, using related ideas, we give game characterizations of Baire measurable and Lebesgue measurable functions.

Games characterizing certain families of functions

Abstract

We obtain several game characterizations of Baire 1 functions between Polish spaces X, Y which extends the recent result of V. Kiss. Then we propose similar characterizations for equi-Bare 1 families of functions. Also, using related ideas, we give game characterizations of Baire measurable and Lebesgue measurable functions.

Paper Structure

This paper contains 5 sections, 13 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Game characterizations of Baire 1 functions
  3. The games $G'_f$ and $G"_f$
  4. Games for equi-Baire 1 families of functions
  5. Game characterization of measurable functions

Key Result

Lemma \oldthetheorem

Let $G(f)$ be a game with a parameter function $f\in Y^X$. For a given class of functions $\mathcal{F}\subset Y^X$ assume that: Then the game $G(f)$ is determined and the class $\mathcal{F}$ can be characterized by $G_f$:

Theorems & Definitions (26)

  • Lemma \oldthetheorem
  • Theorem \oldthetheorem: Le
  • Theorem \oldthetheorem: Ki
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Remark
  • Remark
  • Theorem \oldthetheorem
  • ...and 16 more