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Let (E , T) , C b (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 ∈ C b (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 ⊂ C b (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 C b (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. [ABSTRACT FROM AUTHOR]

Various equicontinuity properties for families of Markov operators have been – and still are – used in the study of existence and uniqueness of invariant probability for these operators, and of asymptotic stability. We prove a general result on equivalence of equicontinuity concepts. It allows comparing results in the literature and switching from one view on equicontinuity to another, which is technically convenient in proofs. More precisely, the characterisation is based on a 'Schur-like property' for measures: if a sequence of finite signed Borel measures on a Polish space is such that it is bounded in total variation norm and such that for each bounded Lipschitz function the sequence of integrals of this function with respect to these measures converges, then the sequence converges in dual bounded Lipschitz norm to a measure. [ABSTRACT FROM AUTHOR]