We work with functions defined on arbitrary subsets of the gen- eralized Baire space $^{cof(κ)}κ$, where κ is an uncountable cardinal satisfying $2^{<κ} = κ$. In this context, we show that the $κ^+$-Borel functions are the smallest collection containing the continuous functions and closed under $≤ κ$-pointwise limits. We introduce the definition of κ-Baire class ξ functions for $1 ≤ ξ < κ^{+}$ and, mirroring the well-known characterization in classical descriptive set theory, we highlight the link between κ-Baire class ξ functions and $Σ^0_{ξ+1}(κ^{+})- measurable functions. This result follows from a stronger theorem that characterizes Baire class 1 functions as $≤ κ$-pointwise limits of full functions, which are exceptionally simple Lipschitz functions. Finally, as in classical descriptive set theory, we characterize with games the continuous functions and the Baire class 1 functions.

This is joint work with Luca Motto Ros.