ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

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Counting measure represents finite-support summation on a discrete LCH space

Example

If $X$ is discrete, every $f \in C_{c} (X)$ has finite support and $L (f) = \sum_{x \in X} f (x)$ is represented by counting measure.

Facts & Assumptions

Given: $X$ has the discrete topology.

Verification

technique · direct

1.1

Compact subsets of a discrete space are finite, so $C_{c} (X)$ consists exactly of finite-support functions. The sum defining $L$ is therefore finite; it is linear and positive.

given

2.1

For counting measure $#$ , integration of a finite-support function is its finite sum, so $L (f) = \int f d #$ . Counting measure is Radon on a discrete space: compact sets are finite and every set is open and is the union of its finite subsets. Its total mass may be infinite when $X$ is infinite.

given∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

24 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

Donald L. Cohn, Measure Theory, 2nd ed., Chapter 7 (standard reference, not scraped)