Axis-parallel rectangles in ${\mathbb{R}}^m$ and their volume

Depends on

• $\mathbb{R}^n$ as the set of functions $n \to \mathbb{R}$, and $d_1$, $d_2$, $d_\infty$ are metrics on it
• The $p$-norms $\lVert x\rVert_p$ for rational $p \ge 1$, and $\lVert x\rVert_\infty$
• Finite sums and finite products, by recursion
• Laws of finite sums and finite products
• Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length
• The canonical natural $\iota(n) = n \cdot 1_F$ of a field
• Canonical naturals are positive and strictly increasing

Used by

• At m=1, cube-nullity and cube-content-zero are exactly the published interval-cover notions Corollary
• At m=1, nondegenerate multidimensional rectangles, grid sums and the integral are exactly the published one-dimensional notions Corollary
• The Jordan content of the parallelepiped spanned by the columns of a square real matrix is the absolute value of its determinant Corollary
• A linear bijection need not preserve Jordan content Counterexample
• Grid partitions of a rectangle in ℝᵐ, their cells, refinements and mesh Definition
• Half-open boxes in ℝⁿ and their volume Definition
• Jordan inner and outer content and Jordan measurable bounded sets in ℝᵐ Definition
• Lower and upper Darboux sums over a grid partition in ℝᵐ Definition
• Measure zero and content zero in ℝᵐ by countable and finite cube covers Definition
• Sections, lower and upper section integrals, and iterated Riemann integrals on product rectangles and Jordan sets Definition
• Tagged grid partitions and Riemann sums in ℝᵐ Definition
• The Riemann integral of a bounded function over a bounded Jordan measurable set Definition
• The support of a function on ℝⁿ and its compactly supported Riemann integral Definition
• A square-integrable kernel without a continuous representative Example
• The Cantor slab C×[0,1] has content zero in ℝ² Example
• The right triangle {(x,y)∈[0,1]²:x+y≤1} has Jordan content 1/2 Example
• The unit box in ℝᵐ has volume 1, and the integral of a constant c over it is c Example
• FALSE: every bounded plane set has Jordan area False statement
• A convex function is bounded above and below on a smaller interior cube Lemma
• A finite rectangle cover admits grid control with arbitrarily small volume excess Lemma
• Countable covers by closed boxes, by open boxes and by closed cubes all compute Lebesgue outer measure Lemma
• Every elementary set is squeezed in volume between a compact subset and an elementary set whose interior contains it Lemma
• For compact subsets of ℝᵐ, measure zero and content zero coincide Lemma
• If every finite interval cover of A⊆ℝ has total length at least c, then every rectangle cover of A×[0,d] has total area at least cd Lemma
• Refinement raises multidimensional lower sums and lowers upper sums, with a quantitative boundary-slab estimate Lemma
• The plane Gaussian integral equals π by polar coordinates 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
• A box with a degenerate side is Lebesgue null, and so is every coordinate hyperplane in ℝⁿ Proposition
• Every at most countable subset of ℝⁿ is Lebesgue null; in particular λ₁(ℚ)=0 Proposition
• A bounded set is Jordan measurable iff its indicator is Riemann integrable, and the integral is its Jordan content Theorem
• A box in ℝⁿ with parameters aᵢ≤ bᵢ is Lebesgue measurable of measure ∏_i<n(bᵢ-aᵢ), whichever of its faces are included Theorem
• A Lipschitz map ℝᵐ→ℝᵐ sends null sets to null sets Theorem
• A subset of ℝ has Lebesgue outer measure zero if and only if it has measure zero in the sense of countable closed-interval covers 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 continuous function on a closed nondegenerate rectangle in ℝᵐ is Riemann integrable Theorem
• Every open subset of ℝⁿ admits a compact Jordan exhaustion Theorem
• Every smooth manifold embeds in some finite-dimensional Euclidean space Theorem
• For a nonzero real c, dilation by c multiplies Lebesgue outer measure by |c|ⁿ, and reflection in the origin preserves it Theorem
• Jordan inner content, outer content, measurability, and content are translation invariant Theorem

…and 4 more results.

Dependency tree · two levels

40 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)