On Borel sets in function spaces with the weak topology
by
Roman Pol
Warsaw University and Miami University
Coauthors: Dennis Burke (Miami University)

New results presented in this talk were obtained by Dennis Burke and myself.

Let (C(K), weak), (C(K)^*, weak^*) be respectively, the space of continuous real-valued functions on a compact space K, and its dual space, equipped with the weak and the weak^* topologies.

Let e:K×(C(K), weak) --> R , e(x, f) = f(x), be the evaluation map .

It turns out that for the class of compact F-spaces, which includes the the Stone-Cech compactification \betaN and its compact subspaces, the evaluation map is not Borel-measurable. In fact, the evaluation is not even C-measurable, i.e., it is not measurable with respect to the smallest \sigma-algebra containing Borel sets and closed under the Souslin operation.

In particular, for l^\infty = C(\betaN), one obtains:

Theorem. The duality map < ,  >:(l^\infty, weak)×((l^\infty)^*, weak^* ) --> R, <x, x^*> = x^*(x), is not C-measurable.

Talagrand proved that norm-discrete sets in l^\infty may not be Borel in the weak topology. Using certain results of Haydon concerning function spaces on trees, one can show that there is a compact scattered space K with Borel-measurable duality map < ,  >: (C(K), weak )×(C(K)^*, weak^* ) --> R, such that certain norm-discrete sets in the space C(K) are not even C-sets in the weak topology.

We shall discuss the phenomenon of non- measurability of the evaluation mapping in connection with some results in the literature concerning the Borel structure of the spaces (C(K), weak) and mappings from Baire spaces into (C(K), weak).

Some essential ideas underlying our proofs are based on the work by Jayne, Namioka and Rogers concerning \sigma-fragmentability of the spaces C(K).

Date received: February 18, 2002