DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-27

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.

Nonnegative simple measurable functions

Definition

Let $(X, A, μ)$ be a measure space. A measurable function $s : X \to [0, + \infty) \subseteq \overline{R}$ (Extended-real-valued measurable functions) is a nonnegative simple measurable function when it has finite range.

Equivalently, there are pairwise disjoint measurable sets $E_{1}, \dots, E_{m} \in A$ and coefficients $c_{1}, \dots, c_{m} \in [0, + \infty)$ such that $s = \sum_{j = 1}^{m} c_{j} χ_{E_{j}} .$ Any such display is a simple representation of $s$ .

Nonnegative simple measurable functions

Definition

Definition references

Depends on

Dependency tree · 2 levels

Sources