LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-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 finite-population design induces probability laws on observed records

Statement

For every fixed $y \in R^{U}$ , the set function $P_{y}^{d}$ defined in Finite populations and sampling designs is a probability measure on $(X_{d}, A_{d})$ .

Facts & Assumptions

Given: A finite nonempty population index set $U$ , a finite sampling support $S_{d}$ with positive weights $p_{d} (s)$ summing to $1$ , and a fixed value function $y : U \to R$ .

[F1]

Finite populations and sampling designs: The observed-record space is $X_{d} = {(s, z) : s \in S_{d}, z : s \to R}$ , its event family is the full power set, and $P_{y}^{d} (B) = \sum_{s : O_{y} (s) \in B} p_{d} (s)$ .

[F2]

Sigma-algebras: A sigma-algebra is an algebra of subsets closed under countable unions.

[F3]

Measures on sigma-algebras: A measure assigns zero to the empty set and is countably additive on every pairwise disjoint sequence of measurable sets.

[F4]

Probability measures and probability spaces: A probability measure is a measure whose total mass is $1$ .

Proof

1.1F1F2algebra

Since $A_{d}$ is the full power set of $X_{d}$ , it contains every subset of $X_{d}$ and is closed under complements and countable unions; hence it is a sigma-algebra by [F2].

1.2F1F4algebra

For every $B \in A_{d}$ , the sum defining $P_{y}^{d} (B)$ is finite and nonnegative. It gives $P_{y}^{d} (\emptyset) = 0$ and $P_{y}^{d} (X_{d}) = \sum_{s \in S_{d}} p_{d} (s) = 1$ .

2.1F1F3F4step 1.2algebra∎

Let $(B_{k})_{k \geq 0}$ be pairwise disjoint in $A_{d}$ and put $C_{k} = {s \in S_{d} : O_{y} (s) \in B_{k}}$ . The sets $C_{k}$ are pairwise disjoint and at most $∣ S_{d} ∣$ of them are nonempty. Their union is ${s : O_{y} (s) \in ⋃_{k} B_{k}}$ , so rearranging these finite weighted sums gives $P_{y}^{d} (⋃_{k} B_{k}) = \sum_{k} P_{y}^{d} (B_{k})$ . By [F3], $P_{y}^{d}$ is a measure; by [F4] and step 1.2, its total mass is $1$ , so it is a probability measure.

Depends on

Used by

Cited to discharge well-definedness by Finite populations and sampling designs.

Dependency tree · two levels

16 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

Shalabh, Sampling Theory, Chapter 2: Simple Random Sampling, IIT Kanpur (standard reference, not scraped)
John J. Borkowski, Horvitz–Thompson Estimators, STAT 446 Lecture 6, Montana State University (standard reference, not scraped)