ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge 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.

A Dirac probability measure concentrates all mass at one point

Example

Let $x_{0} \in X$ . The Dirac probability measure satisfies $δ_{x_{0}} (E) = 1$ exactly when $x_{0} \in E$ and otherwise has value $0$ . For measurable $F$ , its same-ambient restriction satisfies

$(δ_{x_{0}})_{F} = {\begin{cases} δ_{x_{0}}, & x_{0} \in F, \\ 0, & x_{0} \notin F . \end{cases}$

Facts & Assumptions

Given: A measurable space $(X, A)$ , a point $x_{0} \in X$ , and a measurable set $F$ .

[L1]

The Dirac set function at $x_{0}$ is a probability measure and has value $1$ exactly on sets containing $x_{0}$ (A Dirac set function is a probability measure).

[L2]

Restriction to $F$ is the measure $A \mapsto μ (A \cap F)$ on the original sigma-algebra (The restriction of a measure to a measurable set is a measure).

Verification

technique · direct

1.1givenL1

For every measurable $E$ , [L1] gives $δ_{x_{0}} (E) = 1$ if $x_{0} \in E$ and $0$ otherwise; in particular its values on $\emptyset$ and $X$ are $0$ and $1$ .

1.2givenL1L2

By [L2], $(δ_{x_{0}})_{F} (A) = δ_{x_{0}} (A \cap F)$ . If $x_{0} \in F$ , this equals $1$ exactly when $x_{0} \in A$ , so it is $δ_{x_{0}} (A)$ ; if $x_{0} \notin F$ , it is always $0$ .

2.1step 1.1step 1.2∎

Steps 1.1 and 1.2 verify the concentration and both restriction cases, including $F = \emptyset$ and $F = X$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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)