Alphabeta Math
CounterexampleConstruction: AI-generatedVerification: 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.

A version can fail a pointwise identity on a null set

Statement refuted

The assertion “every version of E[0∣F] equals zero at every sample point” is false, under the usual AC conditional-class convention.

Facts & Assumptions

Given: The universal pointwise claim in Statement refuted; a witness will be constructed on two atoms.

[F1]

A real measurable integrable function with the required event integrals is a version of the class. (Conditional expectation as an ae class)

Counterexample

technique · direct
1.1

Let Ω={a,b}, F=P(Ω) and P(A)=1{b∈A}. This is a probability measure: P(Omega)=1, P(empty)=0, and in a disjoint sequence at most one event contains b, so countable additivity holds. Take T=1{a}. It is F-measurable and E∣T∣=1⋅0+0⋅1=0.

given
2.1

For every A subset Omega, ∫AT=T(b)1{b∈A}=0=∫A0. Thus [F1] makes T a version of the conditional expectation of zero. But T(a)=1, so the claimed pointwise identity fails at a. The discrepancy set {a} has probability zero, consistent with almost-sure uniqueness.

step 1.1F1∎

Source notes

Durrett §4.1 uniqueness/version discussion, printed p.206; van der Vaart warning after Lemma 1.10, printed p.4. The two-atom witness is locally constructed.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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