CounterexampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

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An infinite-dimensional Gaussian has no locally finite translation-invariant density reference

Statement refuted

Every centered Borel Gaussian probability on an infinite-dimensional separable Banach space has a density with respect to some nonzero translation-invariant Borel measure that is finite on every bounded Borel set.

Statement references

small-ball MAP comparisons use posterior ball probabilities and do not require pointwise density values

Facts & Assumptions

Given: AC, countable choice, the real Hilbert space $ℓ^{2}$ with standard basis $(e_{j})$ , independent standard normal variables $(ξ_{j})$ , and the displayed series.

[A1]

AC and countable choice are assumed by the Gaussian branch of the weighted Hilbert Gaussian-series result. The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )

[F1]

If $\sum_{j} λ_{j} w_{j} < \infty$ , then $\sum_{j} \sqrt{λ_{j}} ξ_{j} e_{j}$ converges almost surely in the selected Hilbert norm; with standard normal coefficients and AC, its limit is a centered Borel Gaussian random element. Weighted Hilbert Gaussian series and prior regularity

[F2]

A measure is countably additive on pairwise-disjoint measurable sequences, and the measure of a countable union is at most the sum of the measures. Measures on sigma-algebras, Finite and countable subadditivity of measures

[F3]

The rationals are countable and dense in the reals; finite products and countable unions of countable sets are countable. $Q$ is countably infinite, The rationals embed densely in the reals, A product of two at most countable sets is at most countable, Countable unions of at most countable sets, assuming ${A C}_{ω}$

[F4]

A measure is monotone under inclusion of measurable sets. Measures are monotone

Counterexample

Assume AC and countable choice. Let $B = ℓ^{2}$ with its standard basis $(e_{j})_{j \geq 1}$ , let $(ξ_{j})_{j \geq 1}$ be independent standard normal variables, and define $W = \sum_{j \geq 1} \frac{ξ_{j}}{\sqrt{j (j + 1)}} e_{j} .$ The weighted Hilbert Gaussian-series convergence result makes $W$ a centered Borel Gaussian probability on $ℓ^{2}$ . There is no nonzero translation-invariant Borel measure on $ℓ^{2}$ that is finite on every bounded Borel set; here translation invariance means $λ (A + x) = λ (A)$ for every Borel $A \subseteq ℓ^{2}$ and $x \in ℓ^{2}$ . Thus this law has no density with respect to a locally finite translation-invariant analogue of Lebesgue measure.

1.1A1F1givenalgebra

In [F1], take $H_{0} = H_{1} = ℓ^{2}$ , $w_{j} = 1$ , and $λ_{j} = 1 / (j (j + 1))$ . The finite sums telescope as $\sum_{j = 1}^{N} λ_{j} = \sum_{j = 1}^{N} (1 / j - 1 / (j + 1)) = 1 - 1 / (N + 1) \leq 1$ , so the required infinite sum is finite. Thus [F1] applies to the stated independent standard normals and proves that $W$ is a centered Borel Gaussian random element.

1.2F2F4givenalgebra

Suppose that $λ$ is a nonzero locally finite translation-invariant Borel measure, where locally finite means that every point has a neighborhood of finite measure. At zero choose $R > 0$ with $λ (B (0, R)) < \infty$ . Put $r = R / 8$ and $U_{j} = B ((R / 2) e_{j}, r)$ . For $j \neq k$ the distance between the centers is $R / \sqrt{2} > 2 r$ , so these balls are pairwise disjoint, and every $U_{j}$ lies in $B (0, R)$ because $R / 2 + r < R$ . Translation invariance gives a common finite mass $m = λ (U_{j})$ . If $m > 0$ , finite additivity and monotonicity give $N m \leq λ (B (0, R))$ for every $N$ , a contradiction. Hence $m = 0$ . This covers in particular every measure finite on all bounded Borel sets, since that condition implies local finiteness.

2.1F2F3F4step 1.2givenalgebrachoose∎

Translation invariance and step 1.2 make every radius- $r$ ball have measure zero. Let $D$ be the set of finitely supported rational sequences. [F3] makes $D$ countable: it is the countable union, over support lengths $N$ , of the countable set $Q^{N}$ padded with zeros. It is dense in $ℓ^{2}$ : for any $x \in ℓ^{2}$ and $ϵ > 0$ , choose $N$ so that the squared tail sum is less than $ϵ^{2} / 4$ , then approximate the first $N$ coordinates by rationals with squared total error less than $ϵ^{2} / 4$ ; the resulting padded vector is within $ϵ$ of $x$ . The radius- $r$ balls centered at $D$ therefore form a countable cover of $ℓ^{2}$ . By [F2], $λ (ℓ^{2}) = 0$ , and [F4] makes every Borel subset null. This contradicts that $λ$ is nonzero. Hence no such reference measure exists, refuting the statement; in particular, the displayed Gaussian has no density relative to one.

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Sources

M. Dashti, K. J. H. Law, A. M. Stuart, and J. Voss, MAP Estimators and Their Consistency in Bayesian Nonparametric Inverse Problems (standard reference, not scraped)
E. Giné and R. Nickl, Mathematical Foundations of Infinite-Dimensional Statistical Models (standard reference, not scraped)