Measure zero and content zero in ${\mathbb{R}}^m$ by countable and finite cube covers

Depends on

• Axis-parallel rectangles in $\mathbb{R}^m$ and their volume
• Series, partial sums, convergence and the sum, divergence, and the tail series
• A series of nonnegative terms converges iff its partial sums are bounded, and then the sum is their supremum
• Finite sums and finite products, by recursion
• Laws of finite sums and finite products
• Finite, countably infinite, countable, uncountable
• Complete ordered field (least-upper-bound property)

Used by

• A bounded function on a closed bounded interval, or on a closed nondegenerate rectangle, is Riemann integrable exactly when its discontinuity set has Lebesgue measure zero Corollary
• A property holding outside a set of elementary measure zero is exactly a property holding λ-almost everywhere Corollary
• An integrable function whose sections vanish outside finite sets has multiple integral zero Corollary
• At m=1, cube-nullity and cube-content-zero are exactly the published interval-cover notions Corollary
• Jordan content is finitely additive when the overlap has content zero Corollary
• The rational points of [0,1]² form a bounded null set that is not Jordan measurable Counterexample
• Boundary presentations adapted to a simple solid region in a coordinate direction Definition
• Null subsets of a smooth manifold Definition
• A right circular cylinder is an elementary solid region, presented by two caps and four side quarters Example
• The closed ball is an elementary solid region, presented by the eight spherical octants Example
• The closed unit box, with its six faces, is an elementary solid region Example
• The parabola segment {(x,x²):0≤ x≤1} has content zero in ℝ² Example
• A C¹ map sends a compact set of content zero to a set of content zero Lemma
• A finite rectangle cover admits grid control with arbitrarily small volume excess Lemma
• A Lipschitz self-map of ℝⁿ carries Lebesgue null sets to Lebesgue null sets Lemma
• Additivity of the integral over finitely many Jordan pieces that fill a Jordan set up to content zero Lemma
• Change of variables for a C¹ map injective and regular only on the interior of a compact Jordan set Lemma
• Changing a bounded integrand on a content-zero set does not change its Riemann integral Lemma
• Compact null sections imply a compact set is null Lemma
• For compact subsets of ℝᵐ, measure zero and content zero coincide Lemma
• Internal faces cancel and volume integrals add when elementary solid regions are glued Lemma
• Riemann-integrable half-space extensions of chart coefficients Lemma
• Sard on the infinitely flat critical stratum Lemma
• Shared boundary arcs cancel when finitely many elementary regions are glued Lemma
• Subsets and countable unions of null subsets of ℝᵐ are null Lemma
• The boundary of a solid between continuous graphs over a compact Jordan base has content zero Lemma
• The product of a content-zero set and a compact interval has content zero Lemma
• The Riemann integral over a Jordan set is independent of the bounding rectangle Lemma
• Countable unions and subsets of manifold null sets are null Proposition
• Conventions and proved scope for the Riemann integral in ℝᵐ and Jordan content Remark
• A bounded set in ℝᵐ is Jordan measurable iff its boundary is null, equivalently of content zero Theorem
• A Lipschitz map ℝᵐ→ℝᵐ sends null sets to null sets Theorem
• A simple polygon is Jordan measurable and its content is the sum of the contents of its triangles Theorem
• A subset of ℝᵐ has Lebesgue outer measure zero if and only if it is null in the sense of countable closed-cube covers Theorem
• Every smooth manifold embeds in some finite-dimensional Euclidean space Theorem
• Fubini over a bounded Jordan set when all but a content-zero family of sections are integrable Theorem
• Lebesgue outer measure is at most Jordan outer content, and a bounded Jordan measurable set is Lebesgue measurable with Lebesgue measure equal to its Jordan content Theorem
• Lebesgue's criterion in ℝᵐ: a bounded function on a closed nondegenerate rectangle is Riemann integrable iff its discontinuity set is null Theorem
• Riemann--Fubini on product rectangles, with lower and upper section integrals and content-zero exceptional sections Theorem
• The graph of a continuous function on a closed nondegenerate rectangle in ℝᵐ has content zero in ℝᵐ⁺¹ Theorem

…and 1 more result.

Dependency tree · two levels

37 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)