In the case of continuous functions on a compact Hausdorff space, we have that any bounded set is pointwise compact if and only if it is weakly compact and the two topologies coincide with this set. One can prove this by using the sequential compactness criterion for weak compactness (Eberlein-Smulian), and the dominated convergence theorem.
In the case of bounded borel functions on a compact Hausdorff space, is a pointwise compact set necessarily weakly compact?
As the dominated convergence theorem does not hold for the dual of borel functions (finitely additive measures), I was unable to use the same method of proof and am thus unsure.
What additional conditions may give weak compactness in this case?