DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21

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.

Probability measures and probability spaces

Definition

A probability measure on a measurable space $(X, A)$ is a measure $P$ with $P (X) = 1$ (Measures on sigma-algebras). The triple $(X, A, P)$ is a probability space (Measure spaces); $X$ is its sample space and the members of $A$ are its events.

Depends on

Used by

Two four-point probability measures agree on a generating family that is not a pi-system Counterexample
The weights 2⁻⁽ᵏ⁺¹⁾ define a probability measure on P(ℕ) Example
FALSE: agreement on an arbitrary generating family determines a measure False statement
A Dirac set function is a probability measure Proposition

Dependency tree · two levels

5 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. Axler, Measure, Integration & Real Analysis, Definition 12.1 (standard reference, not scraped)