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.

Finitely additive nonnegative set functions

Definition

Let $A$ be a sigma-algebra on $X$ (Sigma-algebras). A finitely additive nonnegative set function is a function $ϕ : A \to [0, + \infty]$ such that $ϕ (\emptyset) = 0$ and

$ϕ (A \cup B) = ϕ (A) + ϕ (B)$

whenever $A, B \in A$ are disjoint. By induction, the same equality holds for every finite pairwise disjoint family, with the empty union and empty sum both equal to $0$ . No countable-additivity condition is part of this definition.

Depends on

Used by

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

G. Folland, Real Analysis, 2nd ed., §1.3 (standard reference, not scraped)