Jaroslav Lukeš
For an open bounded set of
, let the space
consist of all continuous functions on
which are
harmonic on
. Given a continuous function
on the boundary of
denote by
the Dirichlet solution of
.
Further, let
be the space of all bounded functions
on
which are pointwise limits of functions from
.
We show a close relation between some methods of real functions theory and potential theory.
For example, we
indicate proofs that the Dirichlet solution
belongs to the space
.
Moreover, we examine a question whether or not the space
satisfies the ``barycentric formula'' or it is uniformly closed. Solving these problems we use in an essential way the fine topology
methods and Choquet's theory of simplicial spaces.
The situation is quite different when replacing the function space
by the space of continuous affine functions on a compact convex
set and
by the space of Baire-one affine functions
(positive theorems of Choquet and Mokobodzki).
The exposition will be quite elementary and all basic notions will be explained.