Generalized Baire class functions

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 ${\mathbf{\Sigma }}_{\xi +1}^0$-measurable.