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 𝜆+-𝚺^0_{𝜉+1}-measurable.