Vague topology

Let $ X $ be a locally compact Hausdorff space. Assume that $ X $ is second countable (i.e. there is a countable base). Then $ X $ is a Polish space (there exists a complete separable metrization). Let $ \mathfrak X $ be the Borel field of $ X $( cf. Borel field of sets), generated by the (set of open subsets of the) topology of $ X $. Let $ \mathfrak B $ be the ring of all relatively compact elements of $ \mathfrak X $, the ring of bounded Borel sets. Let $ \mathfrak M $ be the collection of all Borel measures on $ X $( cf. Borel measure). Let $ \mathfrak F _ {c} $ be the space of real-valued functions of compact support on $ X $. A sequence of elements $ \mu _ {n} \in \mathfrak M $ converges to $ \mu \in \mathfrak M $ if for all $ f \in \mathfrak F _ {c} $,

$$ \tag{* } \int\limits _ { X } f( x) \mu _ {n} ( dx ) = \ \int\limits _ { X } f( x) \mu ( dx) . $$

The topology thus obtained on $ \mathfrak M $ is called the vague topology. If (*) is required to hold for all bounded continuous functions, one obtains the weak topology on $ \mathfrak M $. Thus, the vague topology is weaker than the weak topology. The difference is illustrated by the observation that a subset $ {\mathcal M} \subset \mathfrak M $ is relatively compact in the vague topology if and only if $ \mu ( B) < \infty $ for all $ \mu \in {\mathcal M} $ and is relatively compact in the weak topology if and only if $ \mu ( X) < \infty $ for all $ \mu \in {\mathcal M} $ and $ \inf _ {B \in \mathfrak B } \sup _ {\mu \in {\mathcal M} } \mu ( X \setminus B) = 0 $.

Let $ \mathfrak N $ be the set of all integer-valued elements of $ \mathfrak M $, i.e. those $ \mu \in \mathfrak M $ for which $ \mu ( B) \in \{ 0, 1, 2, . . . \} $ for all $ B \in \mathfrak B $. Then $ \mathfrak N $ is vaguely closed in $ \mathfrak M $. Both $ \mathfrak N $ and $ \mathfrak M $ are Polish in the vague topology.

If a sequence of real random variables $ Y _ {n} $ on a probability space $ ( \Omega , {\mathcal A} , {\mathsf P} ) $ converges in probability (cf. Convergence in probability) to a random variable $ Y $, then their associated measures converge vaguely. If $ Y $ is $ {\mathsf P} $- almost surely constant, the converse also holds.

References

[a1]	H. Bauer, "Probability theory and elements of measure theory" , Holt, Rinehart & Winston (1972) pp. §7.7 (Translated from German)
[a2]	O. Kallenberg, "Random measures" , Akademie Verlag & Acad. Press (1986) pp. Chapt. 15
[a3]	J. Grandell, "Doubly stochastic Poisson processes" , Springer (1976) pp. Appendix
[a4]	N. Bourbaki, "Intégration" , Eléments de mathématiques , Hermann (1965) pp. Chapt. 1–4, §3.9