Riesz obstruction and the necessity of local compactness for vague convergence

Connection

Riesz’s Lemma (the closed unit ball of a normed space X is compact iff X is finite-dimensional) has a devastating consequence for vague convergence on infinite-dimensional spaces: the test function space C_c(X) collapses to {0}, making vague convergence vacuous. This is not merely a failure of the Riesz-Markov representation theorem — it is a more fundamental obstruction. The entire convergence hierarchy in Vague and weak convergence of signed measures depends on C_c being non-trivial, and Riesz’s Lemma identifies exactly when and why it fails. Hu’s abstract boundedness framework, applied by Basrak-Planinic (2018), provides the natural repair by replacing compact support with support in “bounded” sets — a metric notion that does not suffer from Riesz’s obstruction.

The argument is elementary but sharp. Let X be an infinite-dimensional normed space and suppose f in C_c(X) is not identically zero. Then there exists x_0 with f(x_0) != 0, and by continuity some open ball B(x_0, r) is contained in supp(f). But supp(f) is compact by definition, and compact subsets of infinite-dimensional normed spaces have empty interior (if a compact set contained an open ball, rescaling would show the closed unit ball is compact, contradicting Riesz). This is a contradiction, so f must be identically zero: C_c(X) = {0}.

The consequence for vague convergence (defined as convergence of integrals against all f in C_c(X)) is that mu_n -v→ mu holds trivially for ALL sequences and ALL limits — vague convergence carries zero information. Meanwhile, the stronger notions in the hierarchy (testing against C_0 or C_b) remain meaningful: loose convergence (C_0 test functions) and weak convergence of measures (C_b test functions) are non-trivial even in infinite dimensions.

Bridged Concepts

From Vague and weak convergence of signed measures (Herdegen-Liang-Shelley 2022)

vague convergence: defined via C_c(Omega) — the paper implicitly assumes C_c is non-trivial, which requires local compactness
tightness: Proposition 2.4 (vague + tightness ⇐> weak) becomes vacuous if C_c = {0}, since the vague side is trivially satisfied by all sequences
mass preserving condition: Propositions 2.7, 2.8 use “lim sup |mu_n|(K) ⇐ |mu|(K) for compact K” — on infinite-dimensional spaces, compact sets have empty interior and these conditions become vacuous
The one-point compactification device (Remark 2.6): identifying vague convergence on Omega with weak convergence on Omega_infinity requires Omega to be locally compact for Omega_infinity to be Hausdorff

From A note on vague convergence of measures (Basrak-Planinic 2018) and boundedness - Hu

boundedness - Hu: replaces “compact support” with “support in a bounded set” where B_b is a chosen family of “bounded” sets. Metrically bounded sets (e.g. closed balls of radius R) have non-empty interior in ANY normed space, even infinite-dimensional ones — they are not subject to Riesz’s obstruction
The Hu framework rescues vague convergence from triviality on non-locally-compact spaces: the test functions become bounded continuous functions with support in some B_b-bounded set, which is a non-trivial class

From Infinite Dimensional Analysis (Aliprantis-Border 2006)

Theorem 14.12 / 14.14 (p. 496-497): the Riesz-Markov representation identifying (C_c(X))* with regular signed measures requires X locally compact Hausdorff — this is the representation-theoretic consequence of the same obstruction
Example 14.13 (p. 497): even on compact (but non-metrizable) spaces, C_c can fail to separate measures
Table 14.1 (p. 499): C_c(X) pairs with ca_t(B) only when X is locally compact; C_b(X) pairs with ba_n(A) on any normal space — the C_b duality survives without local compactness

Why It Matters

This observation sharpens the understanding of where the convergence hierarchy in Vague and weak convergence of signed measures lives topologically. The hierarchy

weak ⇒ loose ⇒ vague ⇒ basic ⇒ almost basic

requires C_c != {0} for the bottom three layers to carry information. On infinite-dimensional spaces, the hierarchy collapses to

weak ⇒ loose ⇒ (trivial)

with only weak and loose convergence remaining meaningful. This is not an academic curiosity: measure convergence on infinite-dimensional spaces arises naturally in stochastic PDE theory, statistical mechanics, and the study of Gaussian measures on Banach spaces.

The Hu/Basrak-Planinic framework provides the structural repair, but only for positive measures. Extending the mass preserving condition hierarchy from Vague and weak convergence of signed measures to the Hu setting — replacing compact sets with Hu-bounded sets in the mass-preservation conditions — would fill the “general Polish + signed measures” quadrant:

	Locally compact (C_c non-trivial)	Non-locally-compact (C_c trivial)
Positive measures	Classical (Kallenberg/Resnick)	Hu/Basrak-Planinic
Signed measures	Herdegen-Liang-Shelley	Open

Partial Resolution: Metric Bump Functions Replace Urysohn

A closer analysis reveals that the Herdegen-Liang-Shelley results split into two categories, and the equivalence results extend to the Hu setting while the compactness results do not.

What extends: the mass-preservation hierarchy

The proofs of Propositions 2.4, 2.7, 2.8 use Urysohn’s lemma to construct test functions with compact support. But compact support is unnecessarily strong. On any metric space, one can construct metric bump functions: for any closed set F inside a ball B(x, R), define

phi(y) = max(0, 1 - dist(y, F) / epsilon)

Then phi is continuous, 0 ⇐ phi ⇐ 1, phi = 1 on F, and supp(phi) is contained in F^epsilon — which is metrically bounded (hence B_b-bounded) but NOT necessarily compact. This is the key technical observation: Urysohn constructions can be replaced by metric distance functions throughout.

Define B_b-vague convergence for signed measures: mu_n →^{v,B_b} mu iff integral f dmu_n → integral f dmu for all bounded continuous f with supp(f) in B_b.

Define B_b-mass preservation conditions:

“No mass lost on bounded sets”: lim sup |mu_n|(B) ⇐ |mu|(B) for all B in B_b
“No mass lost globally”: lim sup ||mu_n|| ⇐ ||mu||
“B_b-tightness”: for each eps > 0, exists B in B_b with sup_n |mu_n|(B^c) ⇐ eps

Then the following analogues should hold:

Prop 2.4 analogue: B_b-vague + B_b-tightness ⇐> weak convergence
Prop 2.7 analogue: B_b-vague + no mass lost on bounded sets ⇐> B_b-vague convergence of H-J parts
Prop 2.8 analogue: B_b-vague + lim sup ||mu_n|| ⇐ ||mu|| ⇐> weak convergence of H-J parts

The proofs follow the original structure:

Basrak-Planinic’s portmanteau for positive measures provides the foundation (their Proposition 2.4)
The signed-measure one-sided portmanteau (Theorem A.3 in Herdegen-Liang-Shelley) only needs metric bump functions — the proof constructs phi >= chi_K with phi in C_c; replace this with phi in C_{B_b} via the metric bump construction
C_{B_b} separates signed measures on metric spaces: if integral f dmu = 0 for all f with B_b-bounded support, then mu(G) = 0 for every bounded open set G (approximate chi_G by bump functions), hence mu = 0 since bounded open sets generate the Borel sigma-algebra

What fails: Prokhorov’s theorem

The extension genuinely breaks at Prokhorov’s theorem (Theorem A.4). The classical proof requires:

Tightness concentrates mass on compact sets K
C(K) is separable (K compact metrizable) ⇒ weak* topology is metrizable
Banach-Alaoglu gives weak*-compactness of the unit ball in C(K)*
Diagonal argument extracts convergent subsequences

In the Hu setting, B_b-tightness concentrates mass on metrically bounded sets, which are NOT compact in infinite dimensions (by Riesz’s Lemma). Step 2 collapses: C_b(B(0,R)) is not separable when B(0,R) is not compact, and Banach-Alaoglu gives compactness in a non-metrizable topology where sequences may not suffice.

The sharp dichotomy

Result type	Extends to Hu?	Reason
Equivalences (Props 2.4, 2.7, 2.8): given convergence, what strengthens it?	Yes	Only need metric bumps, not compact support
Compactness (Thm A.4 / Prokhorov): when do convergent subsequences exist?	No	Requires compact sets, killed by Riesz

This means the “general Polish + signed measures” quadrant is partially fillable: the qualitative hierarchy (vague-weak bridge, mass preservation, H-J part convergence) extends, but the compactness theory (relative sequential compactness from tightness) requires additional structure.

Potential Directions

Write out the Hu extension of Propositions 2.4, 2.7, 2.8 for signed measures: The metric bump function replacement is straightforward but needs careful verification at each step. The main check: does the proof of Theorem A.3 (one-sided portmanteau for signed measures) go through with B_b-bounded support instead of compact support?
Recover compactness via Jakubowski’s criterion: Jakubowski (1997) proved a generalisation of Prokhorov’s theorem to non-metrizable spaces, using the concept of a “sequentially determining” family of continuous functions. On a Polish space with Hu boundedness, the proper localising sequence (K_m) might provide such a family. This could rescue Prokhorov’s theorem in the Hu setting without requiring compact sets.
Recover compactness via Aldous-Rebolledo conditions: For stochastic processes, the Aldous-Rebolledo criteria give tightness in spaces where classical Prokhorov fails. If the signed measures arise as laws of processes (e.g. in SPDE applications), these criteria might substitute for the missing compactness.
Extend the Stanek characterisation to Hu-bounded settings: Stanek (2024) showed vague ⇐> basic + local boundedness on R. The sharper question: does this extend to general Polish spaces with Hu boundedness? The distribution function approach needs replacement by a B_b-adapted analogue.
Test case: Gaussian measures on Banach spaces: Weak convergence of Gaussian measures on infinite-dimensional spaces is well-studied (Fernique, Bogachev). Implement the B_b-vague framework for Gaussian measures and check whether the mass-preservation conditions yield new insights about convergence of the positive/negative parts of signed Gaussian-type measures.

Evidence

Riesz’s Lemma: compact sets in infinite-dimensional normed spaces have empty interior (classical, see e.g. Infinite Dimensional Analysis Theorem 6.5 or any functional analysis text)
C_c(X) = {0} on infinite-dimensional normed spaces: follows directly from Riesz’s Lemma plus continuity
Vague and weak convergence of signed measures Theorem 1.2 requires local compactness for Riesz-Markov; Proposition 2.4 uses the one-point compactification (requires local compactness)
boundedness - Hu: Hu’s Corollary 5.12 shows proper localisation works on any Polish space; metrically bounded sets have non-empty interior regardless of dimension
Infinite Dimensional Analysis Theorem 14.14 (p. 497): (C_c(X))* = ca_t(B_X) requires locally compact Hausdorff; Table 14.1 shows C_b(X) duality survives on normal spaces
Metric bump function construction: phi(y) = max(0, 1 - dist(y, F)/eps) is continuous with B_b-bounded support on any metric space — does not require local compactness or Urysohn
C_{B_b} separates signed measures: bounded open sets generate the Borel sigma-algebra, and metric bumps approximate their indicators

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