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+-Σ0ξ+1-measurable.