Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 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.

Integrating against a Dirac measure is evaluation at the point

Example

If δx0 is the Dirac measure at x0, then for every nonnegative measurable f, ∫f dδx0=f(x0), and the same formula holds for every integrable real or complex f.

Facts & Assumptions

Given: A Dirac measure δx0 and a measurable function f.

[L1]

The Dirac set function is a probability measure (The Dirac set function at a point, A Dirac set function is a probability measure).

[L2]

Nonnegative measurable functions admit increasing simple approximations, and monotone convergence passes to the limit of the integrals (Every nonnegative measurable function is the increasing limit of simple measurable functions, Monotone convergence for the integral).

[L3]

Real and complex integrals are defined from the nonnegative theory by positive/negative and real/imaginary parts (Integrable real and complex functions, and their integrals).

Verification

technique · direct
1.1

If s=∑jcjχEj is simple, then [L1, given, algebra] ∫s dδx0=∑jcjδx0(Ej)=s(x0), because exactly one cell containing x0 contributes.

2.1

For nonnegative measurable f, choose simple sn↑f by [L2]. Then [step 1.1, L2, L3] ∎ ∫f dδx0=lim⁡n∫sn dδx0=lim⁡nsn(x0)=f(x0). Apply this to the positive/negative parts and to the real/imaginary parts to obtain the same formula for real and complex integrable f by [L3].

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

20 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