On approximation of affine Baire-one functions

It is known (G. Choquet, G. Mokobodzki) that a Baire-one affine function on a compact convex set satisfies the barycentric formula and can be expressed as a pointwise limit of a sequence of continuous affine functions. Moreover, the space of Baire-one affine functions is uniformly closed. The aim of this paper is to discuss to what extent analogous properties are true in the context of general function spaces. In particular, we investigate the function space H(U), consisting of the functions continuous on the closure of a bounded open set U subset of R-m and harmonic on U, which has been extensively studied in potential theory. We demonstrate that the barycentric formula does not hold for the space B-1(b)(H(U)) of bounded functions which are pointwise limits of functions from the space H(U) and that B-1(b)(H(U)) is not uniformly closed. On the other hand, every Baire-one H(U)-affine function (in particular a solution of the generalized Dirichlet problem for continuous boundary data) is a pointwise limit of a bounded sequence of functions belonging to H(U). It turns out that such a situation always occurs for simplicial spaces whereas it is not the case for general function spaces. The paper provides several characterizations of those Baire-one functions which can be approximated pointwise by bounded sequences of elements of a given function space.