Compact Subsets of the First Baire Class

Perhaps the earliest results about pointwise compact sets of Baire class-1 functions are the two selection theorems of E. Helly found in most of the standard texts on real variable (see, e.g., [Lo], [N]). These two theorems are really theorems about a particular example of a compact set of Baire class-1 functions known today as Helly space, the space of all nondecreasing functions from the unit interval I = [0, 1] into itself. More recently, the notion of Baire class-1 function turned out to also be important in some areas of functional analysis (see [R3]). For example, Odell and Rosenthal [OR] showed that the double dual of a separable Banach space E with the weak∗ topology consists only of Baire class-1 functions defined on the unit ball of E∗ if and only if the space E contains no subspace isomorphic to `. This resulted in a renewed interest in this class of spaces. For example, building on the work of Rosenthal [R2], Bourgain, Fremlin and Talagrand [BFT] proved analogues of the two theorems of Helly for the whole first Baire class. Using their results Godefroy [Go] showed that this class of spaces enjoys some interesting permanence properties. For example, if a compact space K is representable as a compact set of Baire class-1 functions, then so is P (K), the space of all Radon probability measures on K with the weak∗ topology. Some further permanence properties of this class of spaces were obtained by Marciszewski ([M1], [M2]) and an excellent survey of the early results is given by R. Pol [Po2]. Our paper is an attempt towards a fine structure theory of compact subsets of first Baire class. The first result that we give is a positive answer to a natural question one usually asks in such a context.