DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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.

Premeasures on algebras of sets

Definition

Let $A_{0}$ be an algebra of subsets of $X$ . A premeasure on an algebra $A_{0}$ vanishes at the empty set and is countably additive whenever a disjoint sequence in $A_{0}$ has its union in $A_{0}$ . Thus $μ_{0} : A_{0} \to [0, + \infty]$ is a premeasure when $μ_{0} (\emptyset) = 0$ and

$μ_{0} (⋃_{k \in N} A_{k}) = \sum_{k = 0}^{\infty} μ_{0} (A_{k})$

whenever the $A_{k}$ are pairwise disjoint members of $A_{0}$ and their union belongs to $A_{0}$ . Padding a finite disjoint family by empty sets shows that a premeasure is finitely additive.

The premeasure is finite if $μ_{0} (X) < + \infty$ . It is sigma-finite if there is a sequence $(P_{n})$ in $A_{0}$ with $X = ⋃_{n} P_{n}$ and $μ_{0} (P_{n}) < + \infty$ for every $n$ .

Premeasures on algebras of sets

Definition

Depends on

Dependency tree · 2 levels

Sources