Hi there. It is known that on a polish space, if a family of bounded positive measures (no need to be probabilities) is tight, then it is relatively compact in the space of positive measures with usual weak topology. Furthermore, on any metric space, if a family of probability measures is tight, then it is relatively compact in the space of probability measures with the usual weak topology. So the question is, whether it is the case that on a Borel space, if a family of bounded positive measures is tight, then it is relatively compact in the space of positive measures with usual weak topology? (I am aware of some more general results on this about family of signed measures on topological spaces, but let us be specific).

A related question, is the space of positive measures (on Borel spaces endowed with Borel sigma algebra) closed in the space of signed measures equipped with the usual weak topology, i.e. that generated by bounded continuous functions? Many thanks indeed.