Loose convergence is a mode of convergence for sequences of signed measures that sits between weak convergence of measures and vague convergence in the convergence hierarchy. A sequence {mu_n} of finite signed measures on R converges loosely to mu if
integral f d(mu_n) → integral f d(mu) for all f in C_0(R),
where C_0(R) denotes the space of continuous functions vanishing at infinity. This is sometimes also called convergence in the weak-* topology with respect to C_0.
The containment C_c(R) subset C_0(R) subset C_b(R) gives rise to the hierarchy: weak convergence ⇒ loose convergence ⇒ vague convergence
All implications are strict for signed measures. Loose convergence detects mass that is “spread thin” near infinity (unlike vague convergence) but does not detect mass at a single point at infinity (unlike weak convergence).
Stanek (2024) showed in Corollary 3.18 that loose convergence is equivalent to (almost) basic convergence + uniform (not merely local) boundedness in variation: sup_n ||mu_n|| < infinity. This contrasts with vague convergence, which requires only local uniform boundedness. The passage from local to global boundedness is exactly what distinguishes vague from loose convergence.
For non-negative measures, both vague and loose convergence are equivalent since non-negative measures are automatically locally bounded in variation.
Key Details
- Test functions: C_0(R) — continuous functions vanishing at infinity
- Hierarchy: weak ⇒ loose ⇒ vague (all strict for signed measures on R)
- Characterisation: loose convergence ⇐> (almost) basic convergence + uniform boundedness in total variation (Stanek, Corollary 3.18)
- vs vague: the difference is local vs global boundedness in variation
- For non-negative measures: equivalent to vague convergence
- Not metrizable on the space of all finite signed measures on R (Stanek, Corollary 3.20)
- Introduced explicitly by: Stanek (2024) to distinguish from vague convergence for signed measures; some sources define vague convergence using C_0 instead of C_c, making this distinction important