The mass preserving condition is a family of conditions introduced and systematised in Vague and weak convergence of signed measures (Herdegen, Liang, Shelley 2022) that govern the relationship between vague convergence and weak convergence of measures for signed measures. These conditions control whether mass is “lost” in the limiting process.
For a sequence {mu_n} U {mu} in M(Omega) with v-lim mu_n = mu, the key conditions form a hierarchy:
1. Mass not lost at infinity (tightness): For all epsilon > 0, there exists a compact K_epsilon such that lim sup |mu_n|(K_epsilon^c) ⇐ epsilon. This is equivalent to tightness and ensures vague ⇒ weak convergence.
2. Mass not lost on compact sets: lim sup |mu_n|(K) ⇐ |mu|(K) for every compact K in Omega. Combined with vague convergence, this is equivalent to vague convergence of the Hahn-Jordan parts: v-lim mu_n^± = mu^± (Proposition 2.7).
3. Mass not lost globally: lim sup ||mu_n|| ⇐ ||mu|| (where ||.|| is the total variation norm). Combined with vague convergence, this is equivalent to weak convergence of the Hahn-Jordan parts: w-lim mu_n^± = mu^± (Proposition 2.8).
4. Local (zero) mass preserving condition (Definition 3.6): A sequence {mu_n} in M(Omega) has no mass at a point x if for any epsilon > 0, there is an open neighbourhood N_{x,eps} such that lim sup |mu_n|(N_{x,eps}) ⇐ epsilon. The sequence has no mass at +infinity (resp. -infinity) when the canonical extensions satisfy the analogous condition.
This hierarchy is crucial for signed measures because the positive and negative parts can individually lose mass that cancels in the limit. Example 3.5 (mu_n = delta_0 - delta_{1/n}) shows vague convergence to zero with preserved signed mass, but the distribution functions diverge because positive and negative mass cancel locally.
The results are summarised in Table 1 of Herdegen-Liang-Shelley:
- v-lim + tightness ⇒ w-lim (and conversely on Polish spaces)
- v-lim + no compact mass loss ⇐> v-lim of H-J parts
- v-lim + no global mass loss ⇐> w-lim of H-J parts
Key Details
- For positive measures: the only way mass can be lost is at infinity, so tightness suffices for all equivalences
- For signed measures: mass can also be “locally lost” through cancellation of positive and negative parts, requiring the stronger “no mass on compact sets” condition
- On R: the “no mass at any point” condition (no mass at every x in R) is the key additional assumption needed for the equivalence between vague convergence and distribution function convergence (Theorem 3.8)
- Application: these conditions are essential when extending classical results like Karamata’s Tauberian theorem from positive to signed measures
Textbook References
Measure Theory - Bogachev (Bogachev, 2007)
- Theorem 8.4.7 (p. 197): Varadarajan’s theorem — if mu_alpha ⇒ mu weakly, then lim inf |mu_alpha|(U) >= |mu|(U) for all functionally open U. The net |mu_alpha| converges weakly to |mu| precisely when |mu_alpha|(X) → |mu|(X).
- Corollary 8.4.8 (p. 198): If mu_alpha ⇒ mu and |mu_alpha|(X) → |mu|(X), then mu_alpha^+ ⇒ mu^+ and mu_alpha^- ⇒ mu^-. This is the general topological space version of Herdegen-Liang-Shelley’s Proposition 2.8.
- Example 8.4.5 (p. 197): (i) mu_n = sin(nx)dx converges weakly to 0 but |mu_n| has no weak limit; (ii) delta_0 - delta_{1/n} → 0 weakly but |delta_0 - delta_{1/n}| → 2*delta_0. Demonstrates that mass preservation fails for signed measures in general.