CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A nonmeasurable rule is not a statistic or test

Statement refuted

Every set map from a statistical data space to the binary action space is a statistic or a randomized test function.

Facts & Assumptions

Given: The two-point data set, its trivial sigma-algebra, the one-point parameter set, and the map $φ (x) = x$ .

[F1]

A sigma-algebra contains the full space and is closed under complements and countable unions (Sigma-algebras).

[F2]

A measurable space is a set paired with a sigma-algebra (Measurable spaces and measurable sets).

[F3]

The Dirac measure $δ_{0}$ is a probability measure on every sigma-algebra on ${0, 1}$ (The Dirac set function at a point, A Dirac set function is a probability measure).

[F4]

An experiment is a data measurable space, parameter set, and family of probability measures (Statistical experiments).

[F5]

A map is measurable exactly when preimages of measurable target sets are measurable (A measurable function between measurable spaces).

[F6]

A statistic is a measurable data map (Statistics and estimators).

[F7]

A randomized test function is measurable and is the action-1 probability of a binary kernel (Randomized tests).

[F8]

A randomized procedure kernel has measurable action-event evaluations (Randomized procedures as probability kernels).

Counterexample

Given: Let $X = {0, 1}$ carry the trivial sigma-algebra $A = {\emptyset, X}$ . Take a singleton parameter set $Θ = {θ_{0}}$ with sampling law $P_{θ_{0}} = δ_{0}$ , and give the action space ${0, 1}$ its full power-set sigma-algebra. Define the set map $φ (x) = x$ .

1.1givenF1F2

The family $A = {\emptyset, X}$ contains $X$ , and its only members have complements in $A$ ; any countable union of its members is either empty or $X$ . Hence $A$ is a sigma-algebra and $(X, A)$ is a measurable space.

1.2givenF5

In the full action sigma-algebra, ${1}$ is measurable, but $φ^{- 1} ({1}) = {1} \notin A$ . Therefore $φ$ is not measurable by [F5].

2.1givenF3F4step 1.1

The point mass $P_{θ_{0}} = δ_{0}$ is a probability measure on this measurable space by [F3]. Thus the stated one-parameter family is a statistical experiment.

3.1F6F7step 1.2step 2.1

In this valid experiment, the map is not a statistic by [F6] and is not a randomized test function by [F7], since step 1.2 shows it is not measurable.

4.1F8step 1.2step 2.1step 3.1∎

If $φ$ were the action-1 evaluation of a procedure kernel, then $x \mapsto K (x, {1}) = φ (x)$ would have to be measurable by [F8], contradicting step 1.2. Thus the map is not a randomized procedure either, proving the stated claim false.

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