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Vague and weak convergence of signed measures

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Summary

This paper fills a significant gap in the literature by systematically studying weak and vague convergence for signed measures (not just positive measures). For positive measures, the relationship between these convergence modes is classical and well understood. But for signed measures, new phenomena arise because the positive and negative parts in the Hahn-Jordan decomposition can interact in subtle ways — mass from the positive and negative parts can “cancel” in the limit while being individually lost.

The paper’s central organising principle is the hierarchy of “mass preservation” conditions. The key results (summarised in Table 1) show:

  • Vague convergence + tightness ⇒ weak convergence (and conversely on Polish spaces) — this extends to signed measures via Prohorov’s theorem (Theorem A.4 extended)
  • Vague convergence + “no mass lost on compact sets” (lim sup |mu_n|(K) ⇐ |mu|(K) for all compact K) ⇐> vague convergence of Hahn-Jordan parts (v-lim mu_n^± = mu^±)
  • Vague convergence + “no mass lost globally” (lim sup ||mu_n|| ⇐ ||mu||) ⇐> weak convergence of Hahn-Jordan parts (w-lim mu_n^± = mu^±)

For the real line (Section 3), the paper characterises when vague convergence of signed measures is equivalent to pointwise convergence of distribution functions. The classical Helly-Bray type result (Theorem 3.2) — that for positive measures, vague convergence is equivalent to convergence of distribution functions at continuity points — fails for signed measures (Examples 3.4 and 3.5). The new “local (zero) mass preserving condition” (Definition 3.6) — requiring that {mu_n} has no mass at each point — is introduced to rescue the equivalence. Theorem 3.8 gives the definitive characterisation: if the distribution functions converge at continuity points of mu and {mu_n} is bounded on compact sets, then v-lim mu_n = mu; conversely, vague convergence + no mass at continuity points of mu implies distribution function convergence.

The paper was motivated by extending Karamata’s Tauberian theorem to signed measures for applications in stochastic control.

Key Contributions

  • First comprehensive treatment of vague-weak equivalence for signed measures
  • Introduction of the “local (zero) mass preserving condition” (Definition 3.6) to handle signed measure cancellation
  • Table 1 summarising the precise hierarchy: vague + tightness ⇐> weak; vague + no compact mass loss ⇐> vague convergence of H-J parts; vague + no global mass loss ⇐> weak convergence of H-J parts
  • Extension of Prohorov’s theorem to signed measures (Theorem A.4)
  • Theorem 3.8 and 3.11 giving definitive characterisation of vague convergence vs distribution function convergence for signed measures on R
  • Counterexamples (3.4, 3.5, 3.10, 3.12) showing where the positive-measure theory breaks down

Methodology

The paper works on a general metrisable space Omega with the Borel sigma-algebra. Vague convergence is defined via test functions in C_c(Omega) (compact support), weak convergence via C_b(Omega) (bounded continuous). For signed measures, the Hahn-Jordan decomposition mu = mu^+ - mu^- and total variation |mu| = mu^+ + mu^- play a central role. The key technique is using the Riesz-Markov-Kakutani representation theorem (Theorem 1.2) to identify signed measures with continuous linear functionals, and then applying the Stone-Weierstrass theorem (Theorem A.1) to establish density arguments. The vague Portmanteau theorem (Theorem A.2) for positive measures is used as a building block, and Theorem A.3 (extending one direction to signed measures) is proved via Urysohn’s lemma.

Key Findings

  • For signed measures, weak convergence in M(Omega_infinity) is strictly weaker than weak convergence in M(Omega) (Remark 2.6)
  • Tightness exactly lifts vague to weak convergence, even for signed measures (Proposition 2.4)
  • The condition lim sup |mu_n|(K) ⇐ |mu|(K) for compact K is necessary and sufficient for vague convergence to imply convergence of the H-J parts (Proposition 2.7)
  • The condition lim sup ||mu_n|| ⇐ ||mu|| is necessary and sufficient for vague convergence to imply weak convergence of H-J parts (Proposition 2.8)
  • For signed measures on R, vague convergence does NOT imply distribution function convergence at continuity points — a “no mass” condition is needed (Theorem 3.8)
  • Under the no-mass-at-any-point assumption, distribution function convergence and vague convergence become fully equivalent (Theorem 3.11)

Important References

  1. A Tauberian Theorem for signed measures — Herdegen, Liang, Shelley’s companion paper extending Karamata’s theorem to signed measures
  2. A note on vague convergence of measures — Basrak and Planinic’s unified approach to vague convergence
  3. Foundations of modern probability — Kallenberg’s treatment of vague topology and random measures

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