On Baire measurability in spaces of continuous functions

Let C ( K ) be the Banach space of all continuous functions on a given compact space K . We investigate the w ∗ -sequential closure in C ( K ) ∗ of the set of all finitely supported probabilities on K . We discuss the coincidence of the Baire σ -algebras on C ( K ) associated to the weak and pointwise convergence topologies.