The Portmanteau theorem (sometimes attributed to Alexandroff) provides a collection of equivalent characterisations of weak convergence of measures for positive measures. It is one of the most fundamental results in probability theory and measure theory, as it allows one to verify weak convergence through whichever condition is most convenient for a given application.
Classical Portmanteau Theorem (for positive finite measures on a metrisable space Omega): The following are equivalent for {mu_n} U {mu} in M^+(Omega):
- integral f d(mu_n) → integral f d(mu) for all f in C_b(Omega) (weak convergence)
- integral f d(mu_n) → integral f d(mu) for all bounded Lipschitz f
- lim sup mu_n(F) ⇐ mu(F) for every closed set F
- lim inf mu_n(G) >= mu(G) for every open set G
- lim mu_n(A) = mu(A) for every mu-continuity set A (i.e., mu(boundary A) = 0)
There are also vague and unbounded versions:
Vague Portmanteau (for positive measures, locally compact Omega): The following are equivalent:
- v-lim mu_n = mu (vague convergence)
- lim sup mu_n(K) ⇐ mu(K) for compact K, and lim inf mu_n(G) >= mu(G) for open G
- lim mu_n(A) = mu(A) for continuity sets A with compact closure
Portmanteau for unbounded measures (Barczy-Pap 2006): For measures eta_n on a metric space (X, d) that are finite on complements of neighbourhoods of a fixed point x_0 (e.g. Levy measures), six equivalent conditions are given, including convergence of integrals against functions in C(X), C_{x_0}(X), BL_{x_0}(X); weak convergence of restrictions; measure convergence on complements; and lim sup/lim inf conditions on neighbourhoods.
The Portmanteau theorem does NOT directly extend to signed measures in the same form. The direction “(a) ⇒ (b)” of the vague Portmanteau (convergence implies set inequalities) extends to signed measures via Theorem A.3 in Herdegen-Liang-Shelley (attributed to Varadarajan), but the full equivalence requires additional conditions such as mass preservation.
Key Details
- Origin: attributed to A.D. Alexandroff; systematised in Dudley’s treatment
- Key use: verifying weak/vague convergence via the most convenient equivalent condition (often the closed/open set conditions or continuity set condition)
- For distribution functions: the continuity-set condition reduces to convergence of F_n(x) → F(x) at continuity points
- For signed measures: the closed-set/open-set inequalities involve |mu_n| and |mu| (total variation measures), not mu_n and mu directly — partial extensions exist but full equivalence fails
- For unbounded measures: Barczy-Pap corrected errors in Meerschaert-Scheffler and removed separability assumptions
Textbook References
Measure Theory - Bogachev (Bogachev, 2007)
- Theorem 8.2.3 (p. 184): Alexandroff’s portmanteau for probability measures on general topological spaces — uses functionally closed/open sets (not just closed/open). For non-probability nonneg measures, need lim mu_alpha(X) = mu(X) additionally.
- Corollary 8.2.4 (p. 184): On metrizable (or perfectly normal) spaces, functionally closed/open reduces to closed/open. Same if measures are Radon.
- Corollary 8.2.10 (p. 187): Full portmanteau equivalence on metrizable spaces: weak convergence ⇐> closed set upper bound ⇐> open set lower bound ⇐> continuity set convergence
- Theorem 7.10.4 (p. 111): Riesz representation on compact spaces — every continuous linear functional on C(K) is a Radon measure
- Theorem 7.11.3 (p. 116): Riesz representation on locally compact spaces — nonneg linear functionals on C_0(X) represented by Borel measures
Infinite Dimensional Analysis (Aliprantis-Border, 2006)
- Theorem 15.3 (p. 508): Full portmanteau for nets in P(X) on metrizable spaces — 7 equivalent conditions: (1) w*-convergence; (2) integrals against C_b(X); (3) integrals against U_d; (4) integrals against any 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 Borel B with mu(partial B) = 0. Notably uses nets (not just sequences), and includes the U_d characterisation
- Theorem 15.5 (p. 511): Semicontinuous functions induce semicontinuous functionals on P(X) — if f is bounded lsc, mu → integral f dmu is lsc
- Corollary 15.6 (p. 511): {mu : mu(F) >= c} is closed and {mu : mu(V) > c} is open in P(X) for closed F and open V