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

Counting measure on an arbitrary set

Definition

Let $X$ be a set. The counting set function on $P (X)$ is

$#_{X} (E) : = {\begin{cases} ∣ E ∣, & E is finite, \\ + \infty, & E is infinite, \end{cases}$

where finite means equinumerous with a natural number (Finite, countably infinite, countable, uncountable) and $+ \infty$ belongs to the extended real line (The extended real line $\overline{R} = R \cup {- \infty, + \infty}$ , its order, and the arithmetic that is left undefined). The two branches are exhaustive and disjoint. The fact that this set function is a measure, and hence deserves the name counting measure, is proved in Counting measure is a measure ↗.

Depends on

Used by

Dependency tree · two levels

13 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

S. Axler, Measure, Integration & Real Analysis, Example 2.55 (standard reference, not scraped)