Alphabeta Math
DefinitionDefinition: AI-adaptedProof: Not applicableaudited 2026-08-01
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.

Measure zero and content zero in Rm by countable and finite cube covers

Definition

Fix m≥1. A closed cube is a rectangle ∏j<m[aj,aj+ℓ] with ℓ≥0; its volume is ℓm. A set E⊆Rm is null when, for every ε>0, it is covered by a sequence of closed cubes whose nonnegative volume series converges with sum at most ε. It has content zero when such a cover can be finite.

The series and finite sums are Series, partial sums, convergence and the sum, divergence, and the tail series and Finite sums and finite products, by recursion, and their nonnegative bounds use A series of nonnegative terms converges iff its partial sums are bounded, and then the sum is their supremum and Laws of finite sums and finite products. Both properties pass to subsets. Padding a finite cover with degenerate zero-volume cubes proves that content zero implies null. This terminology defines only cover-nullity; it does not define a measure on arbitrary sets.

Depends on

Dependency tree · 2 levels

Sources