Generalized Baire class functions

Abstract

Let $\hat{I}»$ be an uncountable cardinal such that $2^{<\hat{I}»}=\hat{I}»$. Working in the setup of generalized descriptive set theory, we study the structure of $\hat{I}{»}^{+}$-Borel measurable functions with respect to various kinds of limits, and isolate a suitable notion of $\hat{I}»$-Baire class $\hat{I}¾$ function. Among other results, we provide higher analogues of two classical theorems of Lebesgue, Hausdorff, and Banach, namely: (1) A function is $\hat{I}{»}^{+}$-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 $\hat{I}»$. (2) A function is of $\hat{I}»$-Baire class $\hat{I}¾$ if and only if it is $\hat{\mathbf{I}}{\pounds }_{\hat{I}¾+1}^0$-measurable.

33 pages, submitted