Vague convergence is a mode of convergence for sequences of measures that tests against continuous functions with compact support. Given a metrisable space Omega and a sequence {mu_n} of finite signed Radon measures, we say mu_n converges vaguely to mu if
integral f d(mu_n) → integral f d(mu) for all f in C_c(Omega),
where C_c(Omega) denotes the space of continuous real-valued functions with compact support. This is sometimes denoted mu_n -v→ mu.
Vague convergence is strictly weaker than weak convergence of measures: it allows mass to “escape to infinity” because compactly supported test functions cannot detect mass that moves beyond every compact set. The classical example is the sequence of Dirac measures delta_n on R, which converges vaguely to the zero measure (since any compactly supported function eventually vanishes at n) but does not converge weakly.
The relationship between vague and weak convergence is governed by tightness: vague convergence plus tightness is equivalent to weak convergence (Proposition 2.4 in Herdegen-Liang-Shelley). On locally compact spaces, vague convergence can be interpreted as weak convergence on the one-point compactification Omega_infinity.
For positive measures, the Riesz-Markov-Kakutani representation theorem establishes an isometric isomorphism between M(Omega) and (C_c(Omega)), making vague convergence the natural weak- topology. On a locally compact space, when the sequence is uniformly bounded, vague convergence via C_c is equivalent to convergence against C_0 functions (Proposition 1.3 in Herdegen-Liang-Shelley).
For signed measures, new subtleties arise. Mass from the positive and negative parts of the Hahn-Jordan decomposition can cancel in the limit while being individually lost. Additional “mass preserving” conditions are needed to ensure convergence of the parts. Stanek (2024) showed that vague convergence of signed measures on R is equivalent to (almost) basic convergence of distribution functions plus local uniform boundedness in variation.
Basrak and Planinic (2018) showed that different choices of “bounded sets” in Hu’s abstract theory of boundedness yield different flavours of vague convergence — including classical Radon measure vague convergence, the w#-convergence of Daley-Vere-Jones, and the M_0-convergence used in extreme value theory — all as special cases of one unified framework. In each case, the corresponding vague topology is Polish and metrizable.
Key Details
- Test functions: C_c(Omega) — continuous with compact support
- Relation to weak convergence: vague + tightness = weak; vague = weak on compact spaces
- Portmanteau equivalences (positive measures, locally compact): (a) v-lim mu_n = mu; (b) lim sup mu_n(K) ⇐ mu(K) for compact K and lim inf mu_n(Theta) >= mu(Theta) for open Theta; (c) lim mu_n(A) = mu(A) for continuity sets A with compact closure
- For signed measures on R: v-lim mu_n = mu iff {mu_n} is locally bounded in variation and converges basically or almost basically (Stanek’s Theorem 3.12)
- On R (positive measures): vague convergence is equivalent to pointwise convergence of distribution functions at continuity points of the limit (Theorem 3.2, Herdegen-Liang-Shelley)
- On R (signed measures): this equivalence requires additionally that {mu_n} has no mass at continuity points (Theorem 3.8, Herdegen-Liang-Shelley)
- Metrizability: the vague topology on M(X) is metrizable and Polish when the boundedness properly localises X (Basrak-Planinic, Proposition 3.1). But on the space of all finite signed measures on R, vague convergence is NOT metrizable (Stanek, Corollary 3.20)