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.

Weakly convergent sequences are tight

Statement

Assume AC. If μn⇒μ on a Polish space, then {μ,μ1,μ2,…} is tight.

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

Consider any sequence of laws from the displayed family. If some law occurs infinitely often, it has a constant subsequence. Otherwise each law occurs only finitely often. Assign every law unequal to μ its least index in the original sequence. Removing finitely many selected terms for each bounded set of such indices leaves a subsequence whose assigned indices increase to infinity; its weak limit is μ by the given convergence.

givenalgebra
2.1

Thus every sequence in the family has a weakly convergent subsequence with a probability limit on S. The reverse implication of F1, with the stated AC and Polish hypotheses, yields tightness.

F1∎

Depends on

Used by

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Sources