A classical theorem of Kuratowski says that every Baire one function on a Gδ subspace of a Polish (= separable completely metrizable) space X can be extended to a Baire one function on X. Kechris and Louveau introduced a finer gradation of Baire one functions into small Baire classes. A Baire one function f is assigned into a class in this heirarchy depending on its oscillation index β(f). We p...

Abstract We consider a real-valued function f defined on the set of infinite branches X countably branching pruned tree T . The is said to be limsup if there $u \colon \to \mathbb {R}$ such that $f(x) = \limsup _{t \infty } u(x_{0},\dots ,x_{t})$ for each $x \in X$ study game characterization functions, as well novel functions Baire class 1.

In 1990 Kechris and Louveau developed the theory of three very natural ranks on the Baire class 1 functions. A rank is a function assigning countable ordinals to certain objects, typically measuring their complexity. We extend this theory to the case of Baire class ξ functions, and generalize most of the results from the Baire class 1 case. We also show that their assumption of the compactness ...

The investigation of computational properties of discontinuous functions is an important concern in computable analysis. One method to deal with this subject is to consider effective variants of Borel measurable functions. We introduce such a notion of Borel computability for single-valued as well as for multi-valued functions by a direct effectivization of the classical definition. On Baire sp...

We show that small-bound isomorphisms of spaces affine continuous functions on Choquet simplices with Lindel?f boundaries induce Baire measurable bijections their sets extreme points.