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.
At , cube-nullity and cube-content-zero are exactly the published interval-cover notions
Statement
Under , nullity and content zero from cube covers are exactly the published interval-cover notions.
Facts & Assumptions
Given: The standard identification ( as the set of functions , and , , are metrics on it, Each is a norm on , and the induced metrics are exactly , and of the published metric-spaces page).
A one-dimensional closed cube is a closed interval and its volume is its length (Axis-parallel rectangles in and their volume).
Proof
Under the identification, countable cube covers and their volume-series bounds are word for word the countable interval-cover conditions.
The same is true for finite covers and finite sums.
Hence both implications hold for nullity and for content zero.
Depends on
- Measure zero and content zero in $\mathbb{R}^m$ by countable and finite cube covers
- Measure zero (a countable cover by intervals of total length below every $\varepsilon$) and content zero (a finite such cover)
- Axis-parallel rectangles in $\mathbb{R}^m$ and their volume
- $\mathbb{R}^n$ as the set of functions $n \to \mathbb{R}$, and $d_1$, $d_2$, $d_\infty$ are metrics on it
- Each $\lVert\cdot\rVert_p$ is a norm on $\mathbb{R}^n$, and the induced metrics are exactly $d_1$, $d_2$ and $d_\infty$ of the published metric-spaces page
Used by
Dependency tree · two levels
41 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
- J. Lebl, Basic Analysis, Riemann Integral in Several Variables (standard reference, not scraped)
- J. Lebl, Basic Analysis, The Riemann-Lebesgue Criterion (standard reference, not scraped)
- J. Lebl, Basic Analysis, Outer Measure and Null Sets (standard reference, not scraped)