DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-04

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.

Random elements and real random variables

Definition

Let $(Ω, F, P)$ be a probability space and let $(S, Σ)$ be a measurable space. A random element of $S$ is a measurable map $X : (Ω, F) \to (S, Σ)$ in the sense of A measurable function between measurable spaces.

When $S = R$ and $Σ = B (R)$ from The Borel sigma-algebra of a topological space, the map $X$ is a real random variable.

Depends on

Used by

Expectation of a nonnegative or integrable random variable Definition
Law or distribution of a random element Definition
A countable probability space with geometric weights Example
Finite random variables are measurable Lemma
The law of a random element is a probability measure Lemma

Dependency tree · two levels

9 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

S. R. S. Varadhan, Probability Theory, Definition 1.10 (standard reference, not scraped)
J. R. Norris, Probability and Measure, Section 2.1 (standard reference, not scraped)