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.

Finite, sigma-finite, and semifinite measures

Definition

Let $(X, A, μ)$ be a measure space (Measure spaces).

The measure $μ$ is finite if $μ (X) < + \infty$ .
It is sigma-finite if there is a sequence $(E_{n})_{n \in N}$ in $A$ such that $X = ⋃_{n} E_{n}$ and $μ (E_{n}) < + \infty$ for every $n$ .
It is semifinite if, whenever $E \in A$ and $μ (E) > 0$ , there is $F \in A$ with $F \subseteq E$ and $0 < μ (F) < + \infty$ .

The last condition is automatic when $0 < μ (E) < + \infty$ , by taking $F = E$ ; its substantive case is $μ (E) = + \infty$ .

Definition references

Depends on

Dependency tree · 2 levels

Sources

G. Folland, Real Analysis, 2nd ed., §1.3