ExampleConstruction: 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 deterministic statistic is a Dirac kernel

Example

Let $T : (X, A) \to (Y, B)$ be a statistic in an experiment. Define $K (x, C) = δ_{T (x)} (C) = {\begin{cases} 1, & T (x) \in C, \\ 0, & T (x) \notin C, \end{cases} C \in B .$ Then $K$ is a randomized procedure that assigns probability one to the statistic value $T (x)$ after observing $x$ .

Facts & Assumptions

Given: Measurable spaces $(X, A)$ and $(Y, B)$ in a statistical experiment, and a statistic $T : X \to Y$ .

[F1]

A statistic is a measurable map from the observation space to its target measurable space (Statistics and estimators).

[F2]

The Dirac set function at $y$ assigns one to measurable sets containing $y$ and zero to the others (The Dirac set function at a point).

[F3]

Each Dirac set function is a probability measure (A Dirac set function is a probability measure).

[F4]

A randomized procedure is a probability kernel: each action section is a probability measure and each action-event evaluation is measurable in the data (Randomized procedures as probability kernels).

[F5]

A measurable map has measurable preimages (A measurable function between measurable spaces).

[F6]

A sigma-algebra is closed under complements (Sigma-algebras).

Proof

technique · direct

1.1givenF2F3

For each $x \in X$ , the set function $C \mapsto K (x, C)$ is the Dirac measure at $T (x)$ , so it is a probability measure by [F2] and [F3].

1.2F1F5F6

Fix $C \in B$ and put $E = T^{- 1} (C) \in A$ by [F1] and [F5]. The evaluation $x \mapsto K (x, C)$ is $1$ on $E$ and $0$ on $X ∖ E$ , which belongs to $A$ by [F6]. For any Borel subset $D \subseteq [0, 1]$ , its preimage is one of $\emptyset$ , $X$ , $E$ , or $X ∖ E$ , hence is measurable. Thus this evaluation is measurable.

2.1F4step 1.1step 1.2∎

Steps 1.1 and 1.2 verify both clauses of [F4], so $K$ is a probability kernel and therefore a randomized procedure. Since each section is the point mass at $T (x)$ , it takes the action $T (x)$ with probability one.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

15 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.