The theory of boundedness due to Sze-Tsen Hu (1966) provides an abstract framework for defining families of “bounded” sets in a topological space, which in turn determines different notions of locally finite measures and vague convergence. This framework was applied to measure convergence by Basrak and Planinic (2018) to unify several disparate notions of vague convergence in the literature.
A family B_b of subsets of a topological space X is called a (Borel) boundedness if:
- A subset of B in B_b implies A in B_b
- A, B in B_b implies A union B in B_b
- B_b has a basis C_b such that every B in B_b is contained in some C in C_b
A boundedness is proper if it is adapted to a topology (every B in B_b has closure in some open U in B_b). It properly localises X if there exists a metric generating the topology such that metrically bounded sets coincide with B_b.
By Hu’s theorem (Corollary 5.12), B_b properly localises X if and only if there exists a metric generating the topology whose bounded sets are exactly B_b. This permits construction of a proper localising sequence (K_m): a sequence of open bounded sets covering X with nested closures.
The choice of B_b determines everything:
- B_b = all Borel sets: measures finite on all of X ⇒ M^b(X) = finite measures ⇒ vague convergence = weak convergence of measures
- B_b = relatively compact Borel sets (locally compact X): locally finite measures ⇒ classical vague convergence of Radon measures (Kallenberg, Resnick)
- B_b = sets bounded away from a closed C: measures finite away from C ⇒ Hult-Lindskog M_0-convergence used in extreme value theory and regular variation
- B_b = {B: d’(x, C) > epsilon for all x in B, some epsilon > 0}: the M_0-convergence of Lindskog-Resnick-Roy
In each case, vague convergence of mu_n to mu in M(X) means integral f d(mu_n) → integral f d(mu) for all bounded continuous f with support in some set of B_b.
Key Details
- Source: Hu, Introduction to General Topology, 1966
- Key theorem: B_b properly localises X iff it arises from metrically bounded sets of some metric generating X’s topology
- Portmanteau: once B_b properly localises X, the Portmanteau theorem (mu_n(B) → mu(B) for B in B_b with mu(boundary B) = 0) holds automatically via Kallenberg
- Metrizability: the corresponding vague topology on M(X) is metrizable and Polish (Basrak-Planinic, Proposition 3.1)
- Unification: this framework shows that seemingly different convergence notions (classical vague, w#, M_0) are all instances of the same abstract construction with different boundedness families