The Prokhorov metric (sometimes spelled Prohorov) is a metric on the space of finite Borel measures on a separable metric space (X, d) that metrises weak convergence of measures. It was introduced by Yuri Prokhorov in 1956.
For measures mu and nu in M(X), the Prokhorov metric is defined by:
rho(mu, nu) = inf{epsilon > 0 : mu(A) ⇐ nu(A^epsilon) + epsilon and nu(A) ⇐ mu(A^epsilon) + epsilon for all Borel A}
where A^epsilon = {x in X : d(x, A) < epsilon} is the epsilon-enlargement of A. Equivalently, rho(mu, nu) = inf{epsilon > 0 : mu(A) ⇐ nu(A^epsilon) + epsilon for all closed A}.
The key property is that (M^+(X), rho) is a complete separable metric space when X is Polish. For probability measures, rho metrises the topology of weak convergence: mu_n -w→ mu if and only if rho(mu_n, mu) → 0.
Rojas (2021) showed that effective convergence in the Prokhorov metric is equivalent to effective weak convergence for uniformly computable sequences of measures on R (Theorem 4.1). This establishes the Prokhorov metric as the correct computable analogue for studying effective weak convergence.
The Gromov-Prokhorov distance generalises this to metric measure spaces, metrising the Gromov-weak topology used by Foutel-Rodier (2024) for random mmm-spaces.
Key Details
- Metrises: weak convergence of positive finite measures on separable metric spaces
- Completeness: (M^+(X), rho) is Polish when X is Polish
- Prokhorov’s theorem: a family F in M^+(X) is tight iff it is relatively compact in (M^+(X), rho), when X is Polish
- Effective version: effective Prokhorov convergence = effective weak convergence (Rojas 2021)
- Dual representation: rho(mu, nu) = sup{|integral f d(mu) - integral f d(nu)| : f 1-Lipschitz, 0 ⇐ f ⇐ 1} (bounded Lipschitz characterisation)
Textbook References
Measure Theory - Bogachev (Bogachev, 2007)
- Theorem 8.3.2 (p. 193): The Levy-Prohorov metric d_P metrises the weak topology on M_tau^+(X) for any metric space X. On P_tau(X) of probability measures, d_P generates the weak topology. The Kantorovich-Rubinshtein norm ||mu||_0 also metrises on separable spaces.
- Remark (p. 193): If P_sigma(X) != P_tau(X), the weak topology on P_sigma(X) is NOT metrizable — metrizability requires tau-additivity
Infinite Dimensional Analysis (Aliprantis-Border, 2006)
- Theorem 15.11 (p. 513): P(X) is compact and metrizable iff X is compact and metrizable
- Theorem 15.12 (p. 513): P(X) is separable and metrizable iff X is separable and metrizable
- Theorem 15.15 (p. 515): P(X) is Polish iff X is Polish
- Theorem 15.22, Prokhorov (p. 519): On Polish spaces, a nonempty subset of P(X) is relatively compact iff it is tight — the converse direction requires Polish
- Theorem 15.8 (p. 512): x → delta_x embeds X as a closed subset of P(X) when X is separable
- Theorem 15.9 (p. 512): For separable metrizable X, extreme points of P(X) = {delta_x : x in X}