Games characterizing limsup functions and Baire class 1 functions

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 $u:T\to \mathbb{R}$ such that $f(x)={\mathrm{limsup}}_{t\to \infty }u(x_0,\dots ,x_t)$ 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.