Weak and point-wise convergence in C(K) for filter convergence

We study those filters F on N for which weak F-convergence of bounded sequences in C(K) is equivalent to point-wise F-convergence. We show that it is sufficient to require this property only for C[0,1] and that the filter-analogue of the Rainwater extremal test theorem arises from it. There are ultrafilters which do not have this property and under the continuum hypothesis there are ultrafilters which have it. This implies that the validity of the Lebesgue dominated convergence theorem for F-convergence is more restrictive than the property which we study.