CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)

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Noncompact mixtures can have an unattained energy infimum

Statement refuted

For every nonempty Borel generator set $K$ and continuous energy $a : K \to R$ bounded below, the unconstrained infimum of $\int_{K} a d P$ over Borel probability measures on $K$ is attained whenever it is finite, without a compactness or coercivity hypothesis.

Example

Extend the objective in Compact phase-generator mixture to the noncompact Borel generator set $K = (0, 1)$ with energy $a (x) = x$ , and impose no barycenter constraint. The infimum of $\int_{K} x d P (x)$ over Borel probability measures is $0$ , but no probability measure attains it. This shows that a general unconstrained attainment claim needs compactness or a substitute such as coercivity; the example makes no assertion about the barycenter-constrained theorem Attainment and finite support for compact continuous mixtures.

Facts & Assumptions

Given: The explicit noncompact set, continuous energy, and unconstrained Borel-probability objective.

[A1]

$K = (0, 1)$ , $a (x) = x$ , and $P$ ranges over all Borel probability measures on $K$ with objective $\int_{K} x d P$ .

[F1]

Compact phase-generator mixture defines the mean energy $\overset{ˉ}{a} (P) = \int a d P$ ; its constrained mixture energy $a_{m i x}$ is an infimum of those means at a prescribed barycenter. Here [A1] extends only the mean-energy objective to the stated noncompact set and omits the constraint. Monotonicity and nonnegative homogeneity of the nonnegative integral gives monotonicity, and The integral of a nonnegative simple function together with The nonnegative integral agrees with the simple integral on simple functions gives $\int c 1_{E} d P = c P (E)$ for $c \geq 0$ . Since $0 < x < 1$ and $P (K) = 1$ , these rules also give $\int_{K} x d P \leq 1 < \infty$ .

[F2]

Attainment and finite support for compact continuous mixtures: compactness of the generator space is among the hypotheses of its attained-minimum result.

Counterexample

Given: [A1] and the mixture objective [F1].

1.1A1F1algebra

For each integer $n \geq 2$ , the set $E_{n} = [1 / n, 1)$ is Borel, and $⋃_{n \geq 2} E_{n} = (0, 1)$ . Since $P ((0, 1)) = 1$ , at least one $E_{n}$ has positive probability: otherwise the disjoint sets $E_{2}, E_{3} ∖ E_{2}, E_{4} ∖ E_{3}, \dots$ all have probability zero, and countable additivity would give $P ((0, 1)) = 0$ . On $E_{n}$ , $x \geq 1 / n$ , while elsewhere $x \geq 0$ , so $\int_{(0, 1)} x d P \geq \frac{1}{n} P (E_{n}) > 0.$ Therefore every admissible probability has strictly positive objective.

2.1A1F1F2step 1.1∎

The point mass $δ_{1 / n}$ at $1 / n$ is a Borel probability on $(0, 1)$ and has objective $1 / n$ . These values tend to zero, while 1.1 shows all objectives are positive. Hence the infimum is zero and is not attained. The generator set is noncompact, so this conclusion does not contradict the compact result [F2]; it shows why an attainment theorem must retain its compactness/coercivity hypotheses.

Noncompact mixtures can have an unattained energy infimum

Statement refuted

Example

Facts & Assumptions

Counterexample

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