Paper detail

Generalized Baire class functions

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