Point-set games and functions with the hereditary small oscillation property

TL;DR

This paper characterizes families of functions with hereditary oscillation and continuity properties on metric spaces, using point-set games, and explores their measurability and related function classes in Polish spaces.

Contribution

It introduces a game-theoretic framework to characterize hereditary oscillation properties and extends analysis to various measurable and continuous function classes in metric and Polish spaces.

Findings

Characterization of hereditary oscillation property via point-set games

Identification of measurable functions with hereditary properties in Polish spaces

Connections between different classes of functions like cliquish, SZ-functions, and countably continuous functions

Abstract

Given a metric space $X$, we consider certain families of functions $f:X\to\mathbb{R}$ having the hereditary oscillation property HSOP and the hereditary continuous restriction property HCRP on large sets. When $X$ is Polish, among them there are families of Baire measurable functions, $\overline{\mu}$-measurable functions (for a finite nonatomic Borel measure $\mu$ on $X$) and Marczewski measurable functions. We obtain their characterizations using a class of equivalent point-set games. In similar aspects, we study cliquish functions, SZ-functions and countably continuous functions.