DefinitionDefinition: Literature-sourcedProof: Not applicablejudge 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.

Measures on sigma-algebras

Let $X$ be a set and let $A$ be a sigma-algebra on $X$ (Sigma-algebras). A measure on $(X, A)$ is a function

$μ : A ⟶ [0, + \infty] \subseteq \overline{R}$

such that:

$μ (\emptyset) = 0$ ;
for every pairwise disjoint sequence $(E_{k})_{k \in N}$ in $A$ , $μ (⋃_{k \in N} E_{k}) = \sum_{k = 0}^{\infty} μ (E_{k}),$ where the right side is the nonnegative extended sum of Series in the nonnegative extended real line.

The second condition is countable additivity. It begins at index $0$ and includes the case in which some term or the total sum is $+ \infty$ .

Depends on