Tightness is the key condition that bridges vague convergence and weak convergence of measures. It controls the escape of mass to infinity, ensuring that the mass detected by compactly supported test functions accounts for all the mass present.
A sequence {mu_n} of finite signed Radon measures on a metrisable space Omega is called tight if for any epsilon > 0 there exists a compact set K_epsilon such that
sup_n |mu_n|(K_epsilon^c) ⇐ epsilon,
where |mu_n| denotes the total variation measure. Equivalently (by inner regularity), one can replace sup with lim sup.
The central role of tightness is captured by:
Proposition (Herdegen-Liang-Shelley, Prop. 2.4): Let {mu_n} U {mu} be in M(Omega).
- (a) If v-lim mu_n = mu and {mu_n} is tight, then w-lim mu_n = mu.
- (b) If w-lim mu_n = mu, then v-lim mu_n = mu. If Omega is additionally Polish, then {mu_n} is tight.
For signed measures, tightness of {mu_n} means tightness of both {mu_n^+} and {mu_n^-} (the Hahn-Jordan parts). Herdegen-Liang-Shelley prove Prohorov’s theorem for signed measures (Theorem A.4): on a metrisable space, uniform boundedness + tightness implies weak relative sequential compactness.
On R, Herdegen-Liang-Shelley (Remark 3.7) note that tightness of {mu_n} in M(R) is equivalent to the family having no mass at +infinity and no mass at -infinity. This connects tightness to their “mass preserving” conditions.
Stanek (2024) showed that any vaguely convergent sequence of signed measures is automatically locally uniformly bounded in variation (via the uniform boundedness principle). The global tightness condition is what lifts vague convergence to loose or weak convergence.
Key Details
- Definition: for any eps > 0, exists compact K with sup_n |mu_n|(K^c) ⇐ eps
- Core role: tightness + vague convergence ⇐> weak convergence
- Prohorov’s theorem: on a Polish space, {mu_n} is tight iff it is weakly relatively sequentially compact (for uniformly bounded families)
- For probability measures: tightness is automatic when vague convergence holds with a probability limit (by Remark 3.3(b) in Herdegen-Liang-Shelley)
- Helly’s theorem: every sequence of probability measures on R^d has a vaguely convergent subsequence — tightness is exactly the condition ensuring the limit is again a probability measure
- Connection to mass at infinity: on R, tightness ⇐> no mass at +/- infinity ⇐> lim sup ||mu_n|| ⇐ ||mu|| when combined with vague convergence
Textbook References
Measure Theory - Bogachev (Bogachev, 2007)
- Definition 8.6.1 (p. 202): Uniform tightness for Radon and Baire measures — uses |mu|(X \ K_eps) < eps (Radon) or |mu|_*(X \ K_eps) < eps (Baire)
- Theorem 8.6.2 (p. 202): Prohorov’s theorem on complete separable metric spaces — uniform tightness + uniform boundedness ⇐> relative weak sequential compactness
- Theorem 8.6.4 (p. 204): Le Cam’s strengthening — for nonneg Radon measures on any metric space (not necessarily complete), weak convergence implies uniform tightness
- Theorem 8.6.7 (p. 206): Generalisation to completely regular spaces — uniform boundedness + tightness with metrizable compact K_eps implies compact weak closure
- Example 8.6.9 (p. 207): On a countable nonmetrizable space, weakly convergent probability measures may fail to be uniformly tight — shows metrizability is essential
Infinite Dimensional Analysis (Aliprantis-Border, 2006)
- Definition 12.2 (p. 435): A charge or measure mu is tight if mu(A) = sup{mu(K) : K compact, K subset A} for every A; it is regular if every compact set has finite measure and it is both outer regular and tight
- Lemma 12.3 (p. 435): A finite Borel charge is outer regular iff inner regular (iff normal)
- Theorem 12.4 (p. 436): On a Hausdorff space, every tight finite Borel charge is a regular Borel measure (tight charges are automatically countably additive)
- Theorem 12.5 (p. 436): Every finite Borel measure on a metrizable space is normal
- Lemma 12.6 (p. 437): On metrizable spaces, a finite Borel measure is tight iff for each eps > 0 there exists compact K with mu(K) > mu(X) - eps
- Theorem 12.7 (p. 438): Every finite Borel measure on a Polish space is regular (hence tight)
- Definition 15.20 (p. 518): A family F of finite Borel measures on X is tight if for each eps > 0 there exists compact K with mu(K) > mu(X) - eps for each mu in F
- Lemma 15.21 (p. 518): Every tight family of measures in P(X) is relatively compact (separable metrizable X)
- Theorem 15.22, Prokhorov (p. 519): On a Polish space, a nonempty subset of P(X) is relatively compact iff it is tight