DefinitionDefinition: AI-adaptedProof: Not applicableaudited 2026-08-30

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.

Complex simple functions as finite sums of measurable indicators

Definition

Let $(X, A)$ be a measurable space. A function $s : X \to C$ is a complex simple function when there are pairwise disjoint measurable sets $E_{1}, \dots, E_{m} \in A$ and coefficients $c_{1}, \dots, c_{m} \in C$ such that $s = \sum_{j = 1}^{m} c_{j} 1_{E_{j}} .$

Equivalently, $s$ is measurable and has finite range. The disjoint representation above can always be taken to be the canonical one given by the nonempty level sets of $s$ .

Complex simple functions as finite sums of measurable indicators

Definition

Depends on

Dependency tree · 2 levels

Sources