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Riemann's criterion on a nondegenerate rectangle in : integrability is equivalent to arbitrarily small Darboux gaps
Statement
A bounded on a nondegenerate rectangle is Riemann integrable if and only if, for every , some grid satisfies .
Facts & Assumptions
Given: A bounded function on a nondegenerate rectangle.
The lower and upper integrals are the supremum and infimum in The lower and upper Darboux integrals over a nondegenerate rectangle in .
Near-supremum and near-infimum elements exist (Epsilon characterisation of the supremum, Epsilon characterisation of the infimum).
Proof
If the two integrals equal , choose with and with . A common refinement has gap below .
Conversely, a common refinement shows every lower sum is at most every upper sum, so for every , . Arbitrarily small gaps force the integral difference to be .
Thus the conditions are equivalent.
Depends on
- The lower and upper Darboux integrals over a nondegenerate rectangle in $\mathbb{R}^m$
- Lower and upper Darboux sums over a grid partition in $\mathbb{R}^m$
- Refinement raises multidimensional lower sums and lowers upper sums, with a quantitative boundary-slab estimate
- Grid partitions of a rectangle in $\mathbb{R}^m$, their cells, refinements and mesh
- Epsilon characterisation of the supremum
- Epsilon characterisation of the infimum
Used by
- A nonnegative function can have both iterated integrals zero and no double Riemann integral Counterexample
- A Riemann-integrable Thomae-type function whose x-sections are nonintegrable at every rational height Example
- An integrable function on the unit square with one Dirichlet section and only one defined order of ordinary iteration Example
- Changing a bounded integrand on a content-zero set does not change its Riemann integral Lemma
- A bounded set is Jordan measurable iff its indicator is Riemann integrable, and the integral is its Jordan content Theorem
- Every continuous function on a closed nondegenerate rectangle in ℝᵐ is Riemann integrable Theorem
- Lebesgue's criterion in ℝᵐ: a bounded function on a closed nondegenerate rectangle is Riemann integrable iff its discontinuity set is null Theorem
- Linearity, monotonicity, the absolute-value estimate and coordinate-slice additivity for the Riemann integral in ℝᵐ Theorem
- Riemann--Fubini on product rectangles, with lower and upper section integrals and content-zero exceptional sections Theorem
- The multidimensional Darboux and tagged-mesh definitions of the Riemann integral agree Theorem
Dependency tree · two levels
26 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)