Alphabeta MathHow 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.
The Dirac set function at a point
Definition
Let be a sigma-algebra on a nonempty set (Sigma-algebras) and fix . The Dirac set function at is
The two branches are exhaustive and disjoint. The fact that this set function is a probability measure is proved in A Dirac set function is a probability measure ↗.
Depends on
Used by
- Assuming choice, the completion of the Borel Dirac measure at zero is defined on every subset of the real line Example
- The weights 2⁻⁽ᵏ⁺¹⁾ define a probability measure on P(ℕ) Example
- FALSE: measures are additive on arbitrary countable unions False statement
- A Dirac set function is a probability measure Proposition
- A measure on a finite sigma-algebra is a finite weighted sum over its atoms Theorem
- Every measure on a countable discrete space is its weighted sum of Dirac measures Theorem
Dependency tree · two levels
2 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)