Weak convergence of measures is a mode of convergence for sequences of measures that tests against all bounded continuous functions. Given a metrisable space Omega and a sequence {mu_n} of finite signed Radon measures, we say mu_n converges weakly to mu if
integral f d(mu_n) → integral f d(mu) for all f in C_b(Omega),
where C_b(Omega) denotes the space of all bounded continuous real-valued functions. This is denoted mu_n -w→ mu and is sometimes called narrow convergence (particularly in the European literature; see Bogachev).
Weak convergence is strictly stronger than vague convergence: while vague convergence only tests against compactly supported functions and thus allows mass to escape to infinity, weak convergence uses bounded continuous functions that can detect mass everywhere. The sequence delta_n of Dirac measures at n converges vaguely to 0 but does not converge weakly.
The Prokhorov metric metrises weak convergence on M^+(Omega) (positive finite measures on a separable metric space), making it a Polish space. Rojas (2021) showed that for uniformly computable sequences, effective weak convergence is equivalent to effective convergence in the Prokhorov metric.
The Portmanteau theorem provides multiple equivalent characterisations of weak convergence for positive measures: convergence of integrals against bounded continuous functions; against bounded Lipschitz functions; lim sup mu_n(F) ⇐ mu(F) for closed F; lim inf mu_n(G) >= mu(G) for open G; and lim mu_n(A) = mu(A) for all mu-continuity sets A.
The relationship with vague convergence is governed by tightness: for positive measures, vague convergence + tightness is equivalent to weak convergence. This remains true for signed measures (Proposition 2.4 in Herdegen-Liang-Shelley), where tightness is defined in terms of the total variation measure.
For signed measures, weak convergence in M(Omega) is strictly stronger than weak convergence in M(Omega_infinity), where Omega_infinity is the one-point compactification. The former implies vague convergence, but the converse requires tightness.
On R, for probability measures, weak convergence is equivalent to pointwise convergence of distribution functions at all continuity points of the limit (Helly-Bray theorem / portmanteau). For general signed measures, Stanek (2024) showed that loose convergence (the intermediate notion using C_0 test functions) is equivalent to (almost) basic convergence + uniform boundedness in total variation.
Key Details
- Test functions: C_b(Omega) — bounded continuous
- Metrisation: the Prokhorov metric rho(mu, nu) = inf{eps: mu(A) ⇐ nu(A^eps) + eps for all Borel A} metrises weak convergence for positive measures on separable metric spaces
- Portmanteau equivalences (positive measures): integrals of bounded continuous functions; bounded Lipschitz functions; closed set upper bounds; open set lower bounds; continuity set convergence
- Relation to vague: weak ⇒ vague; weak ⇐> vague + tightness
- For probability measures on R: w-lim mu_n = mu iff F_{mu_n}(x) → F_mu(x) at continuity points of F_mu
- Not metrizable on the space of all finite signed measures on R (Stanek, Theorem 3.1)
- Prohorov’s theorem: on a Polish space, a family of measures is tight iff it is weakly relatively sequentially compact
Textbook References
Measure Theory - Bogachev (Bogachev, 2007)
- Definition 8.1.1 (p. 175): Weak convergence defined as convergence of integrals against all f in C_b(X) for Baire measures on a topological space
- Proposition 8.1.7 (p. 177): Banach-Steinhaus — every weakly convergent sequence is bounded in variation norm
- Proposition 8.1.8 (p. 177): On [a,b], weak convergence of signed measures ⇐> sup_n ||mu_n|| < infinity + subsequential convergence of distribution functions at all but countably many points (precursor to basic convergence)
- Corollary 8.2.10 (p. 187): On metrizable spaces, portmanteau equivalences with closed/open sets (not just functionally closed/open)
- Theorem 8.3.2 (p. 193): Levy-Prohorov metric metrises weak topology on nonneg tau-additive measures; weak topology on P_sigma(X) NOT metrizable if P_sigma != P_tau
- Theorem 8.9.4 (p. 213): M_tau^+(X) with weak topology metrizable iff X metrizable
- Remark (p. 182): Weak topology on signed measures is not metrizable (Exercise 8.10.72)
Infinite Dimensional Analysis (Aliprantis-Border, 2006)
- Theorem 15.1 (p. 506): The set U_d of bounded d-uniformly continuous functions is a total set of linear functionals on P(X), separating probability measures
- Theorem 15.2 (p. 507): For any compatible metric d and uniformly dense subset D of U_d, the topologies sigma(P, C_b) = sigma(P, U_d) = sigma(P, D) all coincide on P(X)
- Theorem 15.3 (p. 508): Portmanteau characterisation — 7 equivalent conditions for w*-convergence of nets in P(X) on metrizable spaces, including: (1) mu_alpha →w* mu; (2) integrals against C_b; (3) integrals against U_d; (4) integrals against uniformly dense D in U_d; (5) lim sup mu_alpha(F) ⇐ mu(F) for closed F; (6) lim inf mu_alpha(G) >= mu(G) for open G; (7) mu_alpha(B) → mu(B) for mu-continuity sets B
- Theorem 15.5 (p. 511): If f is bounded and lower (upper) semicontinuous, then mu → integral f dmu is lower (upper) semicontinuous on P(X)
- Theorem 15.11 (p. 513): X compact metrizable iff P(X) compact metrizable
- Theorem 15.12 (p. 513): X separable metrizable iff P(X) separable metrizable
- Theorem 15.15 (p. 515): X Polish iff P(X) Polish
- Theorem 10.14, Vitali-Hahn-Saks (p. 379): If {mu_n} are finite measures on common sigma-algebra and mu_n(A) converges for each A, then mu(A) = lim mu_n(A) defines a finite measure
- Theorem 10.15, Dieudonne (p. 379): On Borel sets of a Polish space, convergence of mu_n(G) for every open G implies convergence of mu_n(B) for every Borel B, and the limit is a finite Borel measure