Generalized Baire class functions

TL;DR

This paper extends classical descriptive set theory results to uncountable cardinals, characterizing generalized Baire class functions and their measurability through limits and hierarchies.

Contribution

It introduces a notion of $oldsymbol{ ext{Baire}}_\xi$ functions in generalized setting and proves higher analogues of classical theorems relating measurability and limits.

Findings

Characterization of $oldsymbol{ ext{Borel}}^+_\xi$-measurable functions via pointwise $D$-limits.

Equivalence of $oldsymbol{ ext{Baire}}_\xi$ class and $oldsymbol{ ext{ extSigma}}^0_{\xi+1}$-measurability.

Extension of classical theorems to uncountable cardinals.

Abstract

Let $\lambda$ be an uncountable cardinal such that $2^{< \lambda } = \lambda$. Working in the setup of generalized descriptive set theory, we study the structure of $\lambda^+$-Borel measurable functions with respect to various kinds of limits, and isolate a suitable notion of $\lambda$-Baire class $\xi$ function. Among other results, we provide higher analogues of two classical theorems of Lebesgue, Hausdorff, and Banach, namely: (1) A function is $\lambda^+$-Borel measurable if and only if it can be obtained from continuous functions by iteratively applying pointwise $D$-limits, where $D$ varies among directed sets of size at most $\lambda$. (2) A function is of $\lambda$-Baire class $\xi$ if and only if it is $\boldsymbol{\Sigma}^{0}_{\xi+1}$-measurable.