Games characterizing limsup functions and Baire class 1 functions

TL;DR

This paper explores game-theoretic characterizations of limsup functions and Baire class 1 functions defined on infinite branches of trees, providing new insights into their structure and properties.

Contribution

It introduces a novel game characterization for Baire class 1 functions and extends game-theoretic methods to limsup functions on tree branches.

Findings

Game characterization of limsup functions established

New game-based description of Baire class 1 functions proposed

Enhanced understanding of function classes via game theory

Abstract

We consider a real-valued function $f$ defined on the set of infinite branches $X$ of a countably branching pruned tree $T$. The function $f$ is said to be a \textit{limsup function} if there is a function $u \colon T \to \mathbb{R}$ such that $f(x) = \limsup_{t \to \infty} u(x_{0},\dots,x_{t})$ for each $x \in X$. We study a game characterization of limsup functions, as well as a novel game characterization of functions of Baire class 1.