Tree structures associated to a family of functions

The research presented in this paper was motivated by our aim to study a problem due to J. Bourgain [3]. The problem in question concerns the uniform boundedness of the classical separation rank of the elements of a separable compact set of the first Baire class. In the sequel we shall refer to these sets (separable or non-separable) as Rosenthal compacta and we shall denote by α(f) the separation rank of a real-valued function f in B1(X), with X a Polish space. Notice that in [3], Bourgain has provided a positive answer to this problem in the case of K satisfying K = K ∩ C(X)p with X a compact metric space. The key ingredient in Bourgain’s approach is that whenever a sequence of continuous functions pointwise converges to a function f , then the possible discontinuities of the limit function reflect a local `-structure to the sequence (fn)n. More precisely the complexity of this `-structure increases as the complexity of the discontinuities of f does. This fruitful idea was extensively studied by several authors (c.f. [5], [7], [8]) and for an exposition of the related results we refer to [1]. It is worth mentioning that A. S. Kechris and A. Louveau have invented the rank rND(f) which permits the link between the c0-structure of a sequence (fn)n of uniformly bounded continuous functions and the discontinuities of its pointwise limit. Rosenthal’s c0-theorem [11] and the c0-index theorem [2] are consequences of this interaction. Passing to the case where either (fn)n are not continuous or X is a non-compact Polish space, this nice interaction is completely lost. Easy examples show that there exist sequences of continuous functions on R pointwise convergent to zero and in the same time they are equivalent to the ` basis. Also there are sequences (fn)n of Baire-1 functions, equivalent to the summing basis of c0, pointwise convergent to a Baire-2 function. Thus if we wish to preserve the main scheme, invented by Bourgain, namely to pass from the elements of the separable Rosenthal compactum to a well-founded tree related to the dense sequence (fn)n, this has to take into account not only the finite subsets of (fn)n but also the points of the Polish space X. This is the key observation on which we have based our approach. Thus for every D subset of R we associate a tree T (fξ)ξ<θ, a, b ) where (fξ)ξ<θ is a wellordering of D and a < b are reals. The elements of the tree are of the form (u, T ) with u a finite increasing subsequence of (fξ)ξ<θ and T a finite dyadic tree in X,