Alphabeta Math
LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedverified 2026-09-26 (gpt-6-sol)
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.

The almost-sure convergence event is measurable

Statement

For real random variables (Xn) and X on one probability space, the set {ω:Xn(ω)→X(ω)} is an event.

Facts & Assumptions

Given: Real random variables (Xn) and X on a probability space.

[L1]

Real convergence may be tested with positive rational tolerances.

Proof

technique · direct
1.1L1

By [L1], the convergence set has the following countable description. [L1] ⋂r∈Q>0 ⋃N=0∞ ⋂n≥N{∣Xn−X∣<r}.

2.1step 1.1∎

Each set {∣Xn−X∣<r} in the display is measurable because Xn−X is a real random variable and the interval (−r,r) is Borel. Countable unions and intersections preserve measurability. Thus the displayed set, and hence the convergence event, is measurable.

Depends on

Used by

Cited to discharge well-definedness by Almost-sure convergence of real random variables.

Dependency tree · two levels

4 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