Ranks on the Baire class $\xi $ functions

Ranks on the Baire Class Ξ Functions

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 ...

Functions Whose Composition with Baire Class One Functions Are Baire Class One

We study the functions whose composition with Baire class one functions are Baire class one functions. We first prove some characterizations of such functions, then investigate a subclass of such functions which are defined in a natural way.

Array Convergence of Functions of the Rst Baire-class

We show that every array (x(i; j) : 1 i < j < 1) of elements in a point-wise compact subset of the Baire-1 functions on a Polish space, whose iterated pointwise limit lim i lim j x(i; j) exists, is converging Ramsey-uniformly. An array (x(i; j) i<j) in a Hausdorr space T is said to converge Ramsey-uniformly to some x in T , if every subsequence of the positive integers has a further subsequence...

On Baire and Harmonic Functions

We consider two spaces of harmonic functions. First, the space H(U) of functions harmonic on a bounded open subset U of R and continuous to the boundary. Second, the space H0(K) of functions on a compact subset K of R n which can be harmonically extended on some open neighbourhood of K. A bounded open subset U of R is called stable if the space H(U) is equal to the uniform closure of H0(U ). We...

2 Sequences of Baire Class 1 Functions 3