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.
For compact subsets of , measure zero and content zero coincide
Statement
A compact subset of is null if and only if it has content zero.
Facts & Assumptions
Given: A compact .
Content zero implies nullity by finite-cover padding (Measure zero and content zero in by countable and finite cube covers).
Compactness is intrinsic, so every ambient-open cover of a compact subset has a finite subcover (Open cover, subcover, compact metric space, and compact subset of a metric space, A subset of a metric space is open in the subspace metric exactly when it is the trace of an open set of the ambient space, and it is compact as a metric space in its own right exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it, The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement).
Proof
One implication is [L1]. For the other, fix and choose a countable closed-cube cover of with total volume below .
Enlarge the -th cube to a larger closed cube whose interior contains it, choosing the added volume below . The interiors form an open cover and the total volumes of their closed containing cubes are below .
Compactness selects finitely many of those interiors. The corresponding finite family of closed enlarged cubes still covers , and its volume sum is at most the entire nonnegative series, hence below .
Thus has content zero.
Depends on
- Measure zero and content zero in $\mathbb{R}^m$ by countable and finite cube covers
- Heine-Borel in $\mathbb{R}^n$: with the Euclidean metric a subset of $\mathbb{R}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line
- Open cover, subcover, compact metric space, and compact subset of a metric space
- A subset of a metric space is open in the subspace metric exactly when it is the trace of an open set of the ambient space, and it is compact as a metric space in its own right exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it
- The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement
- Axis-parallel rectangles in $\mathbb{R}^m$ and their volume
- For $|r| < 1$, $\sum_{k \ge 0} r^k = 1/(1-r)$, and for $|r| \ge 1$ the series diverges
- Finite sums and finite products, by recursion
- Laws of finite sums and finite products
Used by
- The rational points of [0,1]² form a bounded null set that is not Jordan measurable Counterexample
- A C¹ map sends a compact set of content zero to a set of content zero Lemma
- A bounded set in ℝᵐ is Jordan measurable iff its boundary is null, equivalently of content zero Theorem
- Every smooth manifold embeds in some finite-dimensional Euclidean space Theorem
- Lebesgue's criterion in ℝᵐ: a bounded function on a closed nondegenerate rectangle is Riemann integrable iff its discontinuity set is null Theorem
Dependency tree · two levels
55 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)