Alphabeta Math
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.

1.1

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

F1
2.1

Apply that property to the original sequence itself. It supplies increasing indices nj and a Borel probability μ on S with μnj⇒μ. In particular the limit has mass one and lies on S, as asserted.

givenalgebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources