Abstract Let μ1, μ2, … be a sequence of positive Borel measures on ℝ each of which is supported on a set having no finite limit points. Suppose the sequence μn weakly converges to a Borel measure ν.

Atsuji has internally characterized those metric spaces X for which each real-valued continuous function on X is uniformly continuous as follows: (1) the set X' of limit points of X is compact, and (2 ...