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

A simple function and its canonical representation

Definition

Let $(X, A)$ be a measurable space. A function $s : X \to R$ is simple when it is measurable and its range is finite.

If the distinct values taken by $s$ are $c_{1}, \dots, c_{m}$ , define

$E_{j} : = s^{- 1} ({c_{j}}) (1 \leq j \leq m) .$

Then the measurable sets $E_{1}, \dots, E_{m}$ are pairwise disjoint, their union is $X$ , and

$s = \sum_{j = 1}^{m} c_{j} 1_{E_{j}} .$

This expression is the canonical representation of $s$ : the coefficients are the distinct values of $s$ , and the sets are its level sets.

Depends on

A measurable function between measurable spaces

Used by

Dependency tree · two levels

2 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

Sheldon Axler, Measure, Integration and Real Analysis, Definition 2.88 (standard reference, not scraped)