Alethograph

Hu extension of signed measure convergence to general Polish spaces

8 min read

Connection

The results of Vague and weak convergence of signed measures (Table 1: the mass-preservation hierarchy for signed measures) extend from locally compact Polish spaces to arbitrary Polish spaces equipped with a Hu boundedness, by a single systematic substitution: replace “compact” with “B_b-bounded” and Urysohn’s lemma with metric bump functions. The extension preserves the paper’s structure exactly — same Table 1, same hierarchy, same counterexamples — while strictly enlarging its scope. The compactness theory (Prokhorov’s theorem) is recovered via Jakubowski’s sequential Prokhorov theorem (1997), which requires only property CCSP (a countable separating family of continuous functions), not metrizability. This fills the “general Polish + signed measures” quadrant that Basrak-Planinic (2018) left open when they extended the positive measure theory.

The extension is clean because it requires no new ideas — only the observation that every use of compact support in the original proofs can be replaced by B_b-bounded support via metric distance functions, and the observation that the signed measure Prokhorov theorem reduces to the positive measure case via the Hahn-Jordan decomposition.

The Extension

Setup

Let (X, d) be a Polish space with a proper Hu boundedness B_b that properly localises X (Hu’s Corollary 5.12: equivalently, d can be chosen so that B_b = metrically bounded sets). Let (K_m) be the proper localising sequence: open bounded sets with K_m-bar subset K_{m+1} and X = union K_m.

Define:

  • M_{B_b}(X): signed Borel measures mu with |mu|(B) < infinity for all B in B_b (B_b-locally finite)
  • C_{B_b}(X): bounded continuous functions f with supp(f) in B_b
  • B_b-vague convergence: mu_n →^{v} mu iff integral f dmu_n → integral f dmu for all f in C_{B_b}(X)
  • B_b-tightness: for each eps > 0, exists B in B_b with sup_n |mu_n|(B^c) ⇐ eps

Extended Table 1

For {mu_n} union {mu} in M_{B_b}(X) with B_b-v-lim mu_n = mu:

Condition A+Condition B⇐>Result
B_b-vague convergence+B_b-tightness⇐>weak convergence
B_b-vague convergence+no mass lost on bounded sets⇐>B_b-vague convergence of H-J parts
B_b-vague convergence+lim sup ||mu_n|| ⇐ ||mu||⇐>weak convergence of H-J parts

Where “no mass lost on bounded sets” means: lim sup |mu_n|(B) ⇐ |mu|(B) for all closed B in B_b.

Extended Prokhorov (Theorem A.4 analogue)

If {mu_n} subset M_{B_b}(X) is B_b-tight and uniformly bounded (sup_n ||mu_n|| < infinity), then every subsequence contains a B_b-vaguely convergent sub-subsequence.

Recovering the original paper

When X is locally compact and B_b = relatively compact Borel sets:

  • C_{B_b}(X) = C_c(X)
  • B_b-vague convergence = classical vague convergence
  • B_b-tightness = classical tightness
  • The extended Table 1 reduces to the original Table 1

When B_b = all Borel sets (which properly localises any Polish space using the bounded metric d’ = d/(1+d)):

  • C_{B_b}(X) = C_b(X)
  • B_b-vague convergence = weak convergence
  • B_b-tightness is vacuous (every finite measure is trivially tight)
  • Table 1 collapses: only the mass-preservation conditions remain non-trivial

Proof Strategy

Metric bump functions replace Urysohn

Every proof in the original paper that constructs a function phi in C_c with phi = 1 on a compact set K can be replaced by: for closed B in B_b, define

phi(x) = max(0, 1 - dist(x, B) / epsilon)

Then phi is continuous, 0 ⇐ phi ⇐ 1, phi = 1 on B, and supp(phi) subset B^epsilon which is B_b-bounded (since B_b is closed under taking slightly larger bounded sets, by the proper localisation property). This phi plays the role of the Urysohn function throughout.

Specific replacements:

  • Theorem A.3 (one-sided portmanteau for signed measures): replace “compact K” with “closed B in B_b”, use metric bump instead of Urysohn. The proof structure is identical.
  • Proposition 2.4 (vague + tightness ⇐> weak): replace one-point compactification with the proper localising sequence (K_m). The argument: B_b-vague convergence means convergence of integrals against C_{B_b}; B_b-tightness means mass concentrates on bounded sets; together, for any f in C_b(X), approximate f by f * g_m where g_m are cutoff functions from the localising sequence, then g_m * f in C_{B_b} and the remainder is controlled by tightness.
  • Propositions 2.7, 2.8 (mass preservation): replace “compact K” with “closed B in B_b”. The proof uses: (i) one-sided portmanteau gives lim inf |mu_n|(B) >= |mu|(B) for closed B in B_b; (ii) the mass preservation condition gives lim sup |mu_n|(B) ⇐ |mu|(B); (iii) together: |mu_n|(B) → |mu|(B); (iv) by portmanteau for positive measures (Basrak-Planinic), this gives B_b-vague convergence of |mu_n| to |mu|; (v) since mu_n^+ = (mu_n + |mu_n|)/2, convergence of mu_n and |mu_n| gives convergence of mu_n^+.

Jakubowski replaces classical Prokhorov

The original Theorem A.4 (Prokhorov for signed measures) proves: uniform boundedness + tightness implies relative sequential compactness. The proof reduces to Prokhorov for positive measures applied to {mu_n^+} and {mu_n^-}.

In the Hu setting:

  1. B_b-tightness of {mu_n} implies B_b-tightness of {|mu_n|} (since |mu_n|(B^c) >= |mu_n(B^c)|)
  2. B_b-tightness of {|mu_n|} implies B_b-tightness of {mu_n^+} and {mu_n^-}
  3. Basrak-Planinic proved the B_b-vague topology on positive measures is Polish (Proposition 3.1)
  4. On a Polish space, classical Prokhorov applies: B_b-tight + bounded ⇒ relatively compact
  5. Extract convergent subsequences of mu_n^+ and mu_n^- independently
  6. Diagonal argument gives a common subsequence where both converge
  7. mu_n = mu_n^+ - mu_n^- converges in the B_b-vague topology

Alternatively, step 3-4 can be replaced by Jakubowski’s Theorem 2: the space of signed measures with B_b-vague topology satisfies CCSP (metric bumps on bounded sets provide the separating family), so uniform B_b-tightness gives subsequences with Skorokhod representations directly.

C_{B_b} separates signed measures

A necessary check: C_{B_b}(X) must separate signed measures (otherwise B_b-vague convergence is too weak). On a metric space: if integral f dmu = 0 for all f in C_{B_b}(X), then for any bounded open set G, approximate chi_G by metric bump functions f_n with f_n → chi_G pointwise, f_n in C_{B_b}. By dominated convergence (using |mu| as dominating measure), integral f_n dmu → mu(G) = 0. Since bounded open sets generate the Borel sigma-algebra, mu = 0.

Why It Matters

This extension is significant for three reasons:

1. Applications to infinite-dimensional spaces. Stochastic PDE theory, statistical mechanics, and Gaussian measure theory all involve measures on infinite-dimensional spaces (Banach spaces, Frechet spaces, spaces of distributions). The original paper’s results do not apply because these spaces are not locally compact (by Riesz’s Lemma, C_c = {0}). The Hu extension makes the mass-preservation hierarchy available in these settings.

2. Unification with Basrak-Planinic. The Hu extension for signed measures completes the programme started by Basrak-Planinic for positive measures. The four-quadrant table from the Riesz obstruction idea note is now fully filled:

Locally compactGeneral Polish
Positive measuresClassicalBasrak-Planinic
Signed measuresHerdegen-Liang-ShelleyThis extension

3. Minimal effort, maximal impact. The extension requires no new techniques — only metric bumps (elementary) and the Hahn-Jordan reduction to positive measures (already in the paper). The scope increase is large: from locally compact Polish to all Polish spaces.

Potential Directions

  1. Write the extension as a short companion note (2-3 pages): state the extended Table 1 as a theorem, sketch the proof strategy (metric bumps + H-J reduction), and give one application to an infinite-dimensional setting (e.g., signed measures on a separable Hilbert space with weak topology, connecting to Jakubowski’s submetric framework)

  2. Incorporate into a revised version of the paper: add a Section 2.5 “Extension to general Polish spaces” presenting the Hu formulation, with a remark that the proofs of Propositions 2.4, 2.7, 2.8 go through verbatim with the metric bump substitution

  3. Combine with the Barczy-Pap unbounded extension: extend the theory simultaneously to (a) general Polish spaces and (b) unbounded signed measures, giving a portmanteau and mass-preservation hierarchy for unbounded signed measures on non-locally-compact spaces

Evidence

Suggested Papers

idea

Graph View