Let X be any topological space, and C ( X ) C(X) the space of bounded continuous functions on X. We give a nonstandard characterization of weak convergence of a net of bounded linear functionals on C ( X ) C(X) to a tight Baire measure on X. This characterization applies whether or not the net or the individual functionals in the net are tight. Moreover, the characterization is expressed in terms of the values of an associated net of countably additive measures on all Baire sets of X; no distinguished family, such as the family of continuity sets of the limit, is involved. As a corollary, we obtain a new proof that a tight set of measures is relatively weakly compact.