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Refinement raises multidimensional lower sums and lowers upper sums, with a quantitative boundary-slab estimate
Statement
If refines a grid , then . Moreover, for a fixed grid , there is a constant such that refining any grid of mesh by changes either Darboux sum by at most , where .
Facts & Assumptions
Given: The grids, bounded , and bound .
Darboux sums and iterated cell sums are Lower and upper Darboux sums over a grid partition in and Grid partitions of a rectangle in , their cells, refinements and mesh.
Finite sums split and multiplication distributes (Finite sums and finite products, by recursion, Laws of finite sums and finite products).
Proof
Insert one coordinate hyperplane. Every new cell lies in one old cell, so its infimum is no smaller and its supremum no larger. Splitting the affected coordinate sum proves the four inequalities.
Only fine cells meeting an interior hyperplane of can cross a coarse-cell boundary. For a hyperplane perpendicular to coordinate , those cells lie in a slab of thickness at most ; repeated product distributivity bounds their total volume by .
Iterating over the finitely many inserted hyperplanes and coordinates proves refinement monotonicity.
Sum this bound over the finitely many fixed interior hyperplanes to define . On all other cells refinement changes no coarse bound, while on boundary cells each contribution changes by at most times its volume.
This yields the quantitative estimate and completes both assertions.
Depends on
- Lower and upper Darboux sums over a grid partition in $\mathbb{R}^m$
- Grid partitions of a rectangle in $\mathbb{R}^m$, their cells, refinements and mesh
- Axis-parallel rectangles in $\mathbb{R}^m$ and their volume
- Finite sums and finite products, by recursion
- Laws of finite sums and finite products
- The principle of mathematical induction
- The canonical natural $\iota(n) = n \cdot 1_F$ of a field
Used by
- The lower and upper Darboux integrals over a nondegenerate rectangle in ℝᵐ Definition
- A bounded set is Jordan measurable iff its indicator is Riemann integrable, and the integral is its Jordan content 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
32 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)