DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge 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.

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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 $n$ .

Depends on

Used by

Dependency tree · two levels

11 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., Section 1.4 (standard reference, not scraped)
T. Tao, An Introduction to Measure Theory, Definition 1.7.7 (standard reference, not scraped)