Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck 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.

A single jump generates the Dirac mass at 0

Example

Let

F(x):={0,x<0,1,x≥0.

Then the associated Lebesgue-Stieltjes measure is the Dirac mass at 0.

Facts & Assumptions

Given: The step function F displayed above and its Lebesgue-Stieltjes measure μF.

[L1]

Lebesgue-Stieltjes singleton masses are jumps, and half-open interval values are increments. (Interval formulas and atoms for a Lebesgue-Stieltjes measure)

Verification

technique · direct
1.1givenL1

The only jump of F occurs at 0, where F(0)−F(0−)=1, so [L1] gives μF({0})=1.

2.1step 1.1L1∎

If x≠0, then F(x)=F(x−), so [L1] gives μF({x})=0; and for any interval (a,b], the increment F(b)−F(a) is 1 exactly when a<0≤b and is 0 otherwise, which is exactly the interval behavior of δ0.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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