CorollaryStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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.

Tightness extracts a weakly convergent subsequence

Statement

Assume AC. A tight sequence of Borel probability laws on a Polish S has a subsequence converging weakly to a Borel probability on that same S.

Facts & Assumptions

[F1]

Prokhorov tightness theorem on polish spaces: Assume AC. A family $A$ of Borel probabilities on a Polish space S is tight if and only if it is relatively sequentially compact for weak convergence.

Proof

Given: The objects, hypotheses and definitions in the statement. Its conclusions are to be established below.

1.1F1

Let $A = {μ_{n} : n \geq 1}$ . The given uniform compact bounds are exactly tightness of this family. The forward implication of F1 makes it relatively sequentially compact.

2.1givenalgebra∎

Apply that property to the original sequence itself. It supplies increasing indices $n_{j}$ and a Borel probability $μ$ on S with $μ_{n_{j}} \Rightarrow μ$ . In particular the limit has mass one and lies on S, as asserted.

Depends on

Used by

Dependency tree · two levels

13 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

Durrett, §§2.4–2.5, pp. 76–87 (standard reference, not scraped)