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.

Law or distribution of a random element

Definition

Let $X : (Ω, F, P) \to (S, Σ)$ be a random element. Its law or distribution is the set function $P_{X} : Σ \to [0, + \infty], P_{X} (B) : = P (X^{- 1} (B)) .$

Thus the law of $X$ records the probability of each measurable target set by pulling it back to an event in the original probability space.

Depends on

Used by

Cumulative distribution function of a real random variable Definition
A countable probability space with geometric weights Example
Laws commute with measurable maps Lemma
The law of a random element is a probability measure Lemma
Change of variables for expectation Theorem
Probability laws correspond to distribution functions 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

Rick Durrett, Probability: Theory and Examples, 5th ed., Section 1.2 (standard reference, not scraped)
J. R. Norris, Probability and Measure, Section 2.2 (standard reference, not scraped)