Alethograph

Vague and basic convergence of signed measures

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Summary

This paper provides the most complete picture to date of how different convergence notions for signed measures on R relate to each other. It introduces a hierarchy of five convergence types — weak, loose, vague, basic, and almost basic — defined by which class of test functions is used (C_b, C_0, C_c) or by convergence of distribution functions (pointwise vs in Lebesgue measure, up to constants).

The key definitions (Definition 2.1) are:

  • Vague: integrals converge for all f in C_c(R) (compact support)
  • Loose: integrals converge for all f in C_0(R) (vanishing at infinity)
  • Weak: integrals converge for all f in C_b(R) (bounded continuous)
  • Basic (Khartov): every subsequence has a further subsequence along which distribution function differences F(x) - F(y) converge for all x, y outside a countable set
  • Almost basic (new): same as basic but the exceptional set can have Lebesgue measure zero (not just countable)

The hierarchy is: weak ⇒ loose ⇒ vague ⇒ basic ⇒ almost basic. Basic convergence is strictly stronger than almost basic (shown via a Cantor-set construction in Lemma 3.2).

The main result (Theorem 3.12) is a complete characterisation: mu_n converges vaguely to mu if and only if (i) mu_n converges basically or almost basically to mu, AND (ii) (mu_n) is locally uniformly bounded in variation. Moreover, in condition (ii) it suffices that either (mu_n^+) or (mu_n^-) (for any decomposition into non-negative measures) is locally bounded in variation. This generalises results from Herdegen, Liang, and Shelley, who studied pointwise convergence of distribution functions rather than basic/almost basic convergence — the latter being both necessary and sufficient.

The paper also studies metrizability: weak, vague, and loose convergence are not metrizable on the space of all finite signed measures (Theorem 3.1, Corollary 3.20). Almost basic convergence is metrizable via a Ky-Fan-type metric (Lemma 3.4) but the metric space is not complete (Lemma 3.6). Basic convergence is not metrizable (Lemma 3.2).

Key Contributions

  • Introduces almost basic convergence as a natural relaxation of Khartov’s basic convergence
  • Main theorem (3.12): vague convergence ⇐> (almost) basic convergence + local uniform boundedness in variation
  • Shows the two conditions in Theorem 3.12 are independent (Remark 3.17)
  • Proves weak/vague/loose convergence are not metrizable on signed measures (Theorem 3.1, Corollary 3.20)
  • Proves almost basic convergence is metrizable (Lemma 3.4) but basic convergence is not (Lemma 3.2)
  • For non-negative measures, basic and almost basic convergence are equivalent to vague convergence (Remark 3.17(c))
  • Analogous characterisation for loose convergence via uniform (not just local) boundedness in variation (Corollary 3.18)
  • Shows that local boundedness of either the positive or negative parts suffices (improvement on needing both)

Methodology

The paper works on R with the Borel sigma-algebra. Distribution functions are F^(mu)(x) = mu((-inf, x]). Basic convergence is defined via subsequential convergence of differences F(x) - F(y), and almost basic convergence relaxes the exceptional set from countable to Lebesgue-null. The Ky Fan-type metric for almost basic convergence (Lemma 3.4) is d(f,g) = min{eps: lambda({x in [-1/eps, 1/eps]: |f(x) - c - g(x)| > eps} for some c) ⇐ eps}. Theorem 3.12 is proved by showing: basic convergence + local boundedness ⇒ vague convergence (via integration by parts and dominated convergence), and conversely: vague convergence ⇒ local boundedness (via the uniform boundedness principle / Banach-Steinhaus) + basic convergence (via localization with cutoff functions).

Key Findings

  • The hierarchy weak ⇒ loose ⇒ vague ⇒ basic ⇒ almost basic is strict for signed measures
  • For non-negative measures, vague = basic = almost basic (the hierarchy collapses)
  • Vague convergence of signed measures requires TWO conditions: a “shape” condition (basic/almost basic convergence of distribution functions) AND a “size” condition (local uniform boundedness in variation)
  • These two conditions are independent: Example 2.7 shows basic convergence without vague convergence (no local boundedness), and Remark 3.17 shows local boundedness without basic convergence
  • Weak convergence is equivalent to loose convergence + convergence of total masses (follows from Theorem 3.11)
  • The uniform boundedness principle provides an elegant proof that vaguely convergent sequences must be locally bounded in variation

Important References

  1. Vague and weak convergence of signed measures — Herdegen, Liang, Shelley’s treatment relating vague convergence to distribution function convergence
  2. On weak convergence of quasi-infinitely divisible laws — Khartov’s introduction of basic convergence
  3. Weak convergence of measures — Bogachev’s comprehensive monograph

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