On the local compactness of spaces of positive measures

Let (E,T), Cb(E,T) and M be a locally compact Polish space, the set of bounded continuous functions f:E→R and the set of all positive finite Borel measures defined on the corresponding Borel σ-field B(E,T), respectively. Then M equipped with the weak topology defined by means of all f∈Cb(E,T) turns into a Polish space, which fails to be locally compact if (E,T) is not compact. In this note, we explicitly indicate subsets F⊂Cb(E,T) such that M with the topology defined similarly as the weak topology, but with f∈F only, gets locally compact. To this end, by constructing special metrics we introduce coarser topologies, T′, for each of which B(E,T)=B(E,T′) and (E,T′) is compact. Then Cb(E,T′) are taken as the corresponding sets F. An application of this result to measure-valued Markov processes is also provided. Additionally, we show that M endowed with the topology induced from the weak topology of the space of all finite positive measures on the Alexandroff compactification of (E,T) fails to be locally compact. Our technique also allows one to specify metrics which make M a compact metric space.