DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedaudited 2026-09-07

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.

Zero truncation at a positive level

Definition

For a real random variable $X$ and a deterministic level $A > 0$ , its zero truncation is $X^{(A)} = X 1_{{∣ X ∣ \leq A}} .$ The threshold event is measurable because $X$ is measurable and $[- A, A]$ is Borel; its indicator and the product are measurable by Arithmetic and lattice operations preserve measurability whenever they are defined. Thus $X^{(A)}$ is a real random variable as in Random elements and real random variables. It equals $X$ at both cutoff endpoints and is zero outside the interval. Since $∣ X^{(A)} ∣ \leq A$ , for every $0 < p < \infty$ its absolute $p$ th moment is at most $A^{p} P (Ω) = A^{p}$ . This is not clipping to the endpoints.

Depends on

Used by

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

Section 2.2.3, p. 62 (standard reference, not scraped)