LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge 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.

Every measurable rejection function defines a randomized test

Statement

Let $E = ((X, A), Θ, (P_{θ})_{θ \in Θ})$ be a statistical experiment (Statistical experiments) and let $φ : X \to [0, 1]$ be measurable, where $[0, 1]$ has its Borel sigma-algebra. On the binary action space ${0, 1}$ with its full sigma-algebra, define $K (x, B) = φ (x) 1_{B} (1) + (1 - φ (x)) 1_{B} (0) (B \subseteq {0, 1}) .$ Then $K$ is a probability kernel and hence a randomized procedure; its randomized-test function under the convention $1 = r e j e c t$ is $φ$ .

Facts & Assumptions

Given: A statistical experiment $E$ and a measurable function $φ : X \to [0, 1]$ on its data space.

[F0]

Statistical experiments: an experiment supplies a data measurable space and its parameter-indexed probability laws.

[F1]

A measurable function between measurable spaces: measurability means the preimage of every Borel subset of $[0, 1]$ belongs to $A$ .

[F2]

Closure properties of measurable functions used by the integral: sums and real scalar multiples of measurable real functions are measurable.

[F3]

Measure kernel and probability kernel: a probability kernel has probability-measure sections and measurable event evaluations.

[F4]

Sigma-algebras: the full collection of subsets of ${0, 1}$ is a sigma-algebra.

[F5]

Randomized procedures as probability kernels: for the supplied experiment, a probability kernel from the data space to the action space is a randomized procedure.

[F6]

Randomized tests: a binary randomized procedure has test function $x \mapsto K (x, {1})$ .

Proof

technique · direct

1.1givenF0F3F4algebra

Fix $x \in X$ . The two weights $φ (x)$ and $1 - φ (x)$ are nonnegative and sum to one. Thus $K (x, \emptyset) = 0$ and $K (x, {0, 1}) = 1$ . If $(B_{j})_{j \geq 0}$ are pairwise disjoint subsets of ${0, 1}$ , each action belongs to at most one $B_{j}$ , so $1_{\cup_{j} B_{j}} (a) = \sum_{j} 1_{B_{j}} (a)$ for $a = 0, 1$ . Substitution into the formula for $K$ gives $K (x, \cup_{j} B_{j}) = \sum_{j} K (x, B_{j})$ ; hence $K (x, \cdot)$ is a probability measure on the binary action sigma-algebra.

2.1F1F2F3F5F6step 1.1∎

For the four action events, the evaluation functions are $K (x, \emptyset) = 0$ , $K (x, {0}) = 1 - φ (x)$ , $K (x, {1}) = φ (x)$ , and $K (x, {0, 1}) = 1$ . They are measurable by [F1] and [F2]. Therefore [F3] makes $K$ a probability kernel; [F5] makes it a randomized procedure, and [F6] identifies its test function as $K (x, {1}) = φ (x)$ .

Depends on

Used by

Dependency tree · two levels

17 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.

Sources

Ryan Martin, Lecture Notes on Advanced Statistical Theory, §5.1, PDF pp. 100–102 (standard reference, not scraped)
Peter Hoff, Statistical Decision Problems, §§2–4.2, PDF pp. 2–8 (standard reference, not scraped)