GENERALIZED BAIRE CLASS FUNCTIONS

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