Return to Revisions

The proper framework for your question is the so-called strict topology on the space of bounded continuous functions which was introduced for the case of a locally compact space by R.C. Buck in the 50's and extended to the case of a completely regular space by several authors around 1970. It has the property that the dual is the space of tight Radon measures. In your case (locally compact and metric, and so paracompact), there are very strong results available, particularly in the direction that you are interested in---characterisations of compactness and convergence for families of measures). A central result concerns the relationship between weak compactness and uniform tightness of such families. You might start by consulting the article "The stzrict topology and compactness in the space of measures" by John B. Conway which appeared in the Transactions 126 (1967) 474-486.