LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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 stopped random variable is measurable at the stopping time

Statement

If $X$ is adapted, $τ$ is a stopping time, and the value on ${τ = \infty}$ is a fixed real number, then $X_{τ}$ is $F_{τ}$ -measurable. In particular, $X_{τ \land n}$ is $F_{τ \land n}$ -measurable.

Facts & Assumptions

Given: The hypotheses, objects, and conventions in the Statement.

[F1]

Stopped random variable and stopped process gives the disjoint level-set formula.

[F2]

Sigma-algebra at a stopping time gives the event tests for $F_{τ}$ .

[F3]

Equivalent event tests for a discrete stopping time makes ${τ = k} \in F_{k}$ .

Proof

1.1

For Borel $B$ , ${X_{τ} \in B} = ⋃_{k \geq 0} ({τ = k} \cap {X_{k} \in B}) \cup ({τ = \infty} if x_{\infty} \in B) .$ All finite-level terms lie in $F$ , and ${τ = \infty}$ is the complement of their countable union, so the inverse image is in $F$ .

F1F3

2.1

Intersecting that inverse image with ${τ \leq m}$ deletes the infinity term and all levels above $m$ , leaving $⋃_{k = 0}^{m} {τ = k} \cap {X_{k} \in B} \in F_{m} .$ Thus every inverse image passes F2's tests and $X_{τ}$ is $F_{τ}$ -measurable. Apply the same proof to the bounded stopping time $τ \land n$ for the final assertion.

F2F3step 1.1∎

Depends on

Used by

Stopping a likelihood-ratio martingale Example
Optional sampling for bounded stopping times Theorem

Dependency tree · two levels

6 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

van der Vaart, Martingales, Diffusions and Financial Mathematics, Exercise 2.40, p. 20 (standard reference, not scraped)