Alphabeta MathThe Riemann Integral in R^m and Jordan Content
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1 · Prerequisites
Abstract Algebra
Linear Algebra
Real Analysis
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Series: Convergence and the Nonnegative Tests
- Suprema and Infima
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Riemann Integral: Definition and Integrability
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Definition
Axis-parallel rectangles in ℝᵐ and their volume
Definition
Grid partitions of a rectangle in ℝᵐ, their cells, refinements and mesh
Definition
Lower and upper Darboux sums over a grid partition in ℝᵐ
Lemma
Refinement raises multidimensional lower sums and lowers upper sums, with a quantitative boundary-slab estimate
Definition
The lower and upper Darboux integrals over a nondegenerate rectangle in ℝᵐ
Theorem
Riemann's criterion on a nondegenerate rectangle in ℝᵐ: integrability is equivalent to arbitrarily small Darboux gaps
Definition
Tagged grid partitions and Riemann sums in ℝᵐ
Theorem
The multidimensional Darboux and tagged-mesh definitions of the Riemann integral agree
Corollary
At m=1, nondegenerate multidimensional rectangles, grid sums and the integral are exactly the published one-dimensional notions
Theorem
Linearity, monotonicity, the absolute-value estimate and coordinate-slice additivity for the Riemann integral in ℝᵐ
Theorem
Every continuous function on a closed nondegenerate rectangle in ℝᵐ is Riemann integrable
Definition
Measure zero and content zero in ℝᵐ by countable and finite cube covers
Corollary
At m=1, cube-nullity and cube-content-zero are exactly the published interval-cover notions
Lemma
Subsets and countable unions of null subsets of ℝᵐ are null
Lemma
For compact subsets of ℝᵐ, measure zero and content zero coincide
Definition
Oscillation of a real function on subsets of ℝᵐ and at a point
Lemma
A function on a subset of ℝᵐ is continuous at x iff its oscillation there is 0, and every oscillation superlevel set is closed
Lemma
A finite rectangle cover admits grid control with arbitrarily small volume excess
Theorem
Lebesgue's criterion in ℝᵐ: a bounded function on a closed nondegenerate rectangle is Riemann integrable iff its discontinuity set is null
Definition
Jordan inner and outer content and Jordan measurable bounded sets in ℝᵐ
Theorem
A bounded set is Jordan measurable iff its indicator is Riemann integrable, and the integral is its Jordan content
Theorem
A bounded set in ℝᵐ is Jordan measurable iff its boundary is null, equivalently of content zero
Definition
The Riemann integral of a bounded function over a bounded Jordan measurable set
Lemma
The Riemann integral over a Jordan set is independent of the bounding rectangle
Theorem
A continuous real function on a compact Jordan measurable set is Riemann integrable over that set
Corollary
Jordan content is finitely additive when the overlap has content zero
Theorem
A Lipschitz map ℝᵐ→ℝᵐ sends null sets to null sets
Theorem
The graph of a continuous function on a closed nondegenerate rectangle in ℝᵐ has content zero in ℝᵐ⁺¹
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
Remark