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Lower and upper Darboux sums over a grid partition in
Definition
Let be bounded on a nondegenerate rectangle and let be a grid. For each cell , put The extrema exist as finite reals because each nonempty image is bounded (Lower bound, bounded below, bounded set, Complete ordered field (least-upper-bound property), Every nonempty set bounded below has an infimum, Greatest lower bound (infimum), Suprema and infima are unique), and the sums use the iterated convention of Grid partitions of a rectangle in , their cells, refinements and mesh.
Since and cell volumes are nonnegative, . Moreover the sum of cell oscillations weighted by volume (Laws of finite sums and finite products).
Depends on
- Grid partitions of a rectangle in $\mathbb{R}^m$, their cells, refinements and mesh
- Axis-parallel rectangles in $\mathbb{R}^m$ and their volume
- Lower bound, bounded below, bounded set
- Complete ordered field (least-upper-bound property)
- Greatest lower bound (infimum)
- Every nonempty set bounded below has an infimum
- Suprema and infima are unique
- Laws of finite sums and finite products
Used by
- At m=1, nondegenerate multidimensional rectangles, grid sums and the integral are exactly the published one-dimensional notions Corollary
- Tagged grid partitions and Riemann sums in ℝᵐ Definition
- The lower and upper Darboux integrals over a nondegenerate rectangle in ℝᵐ Definition
- The unit box in ℝᵐ has volume 1, and the integral of a constant c over it is c Example
- A product grid bounds the Darboux sums of the lower and upper section-integral functions Lemma
- Borel change of variables from the compact-support formula and Radon uniqueness Lemma
- Borel Darboux integrands in finite dimension Lemma
- Refinement raises multidimensional lower sums and lowers upper sums, with a quantitative boundary-slab estimate 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's criterion on a nondegenerate rectangle in ℝᵐ: integrability is equivalent to arbitrarily small Darboux gaps Theorem
- The multidimensional Darboux and tagged-mesh definitions of the Riemann integral agree Theorem
Dependency tree · two levels
28 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)