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A bounded set is Jordan measurable iff its indicator is Riemann integrable, and the integral is its Jordan content
Statement
A metric-bounded set is Jordan measurable if and only if its indicator is Riemann integrable on a fixed nondegenerate bounding rectangle . In that case
Facts & Assumptions
Given: Metric-bounded in the sense of Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space, where is a nondegenerate bounding rectangle.
Jordan inner and outer content are Jordan inner and outer content and Jordan measurable bounded sets in .
Multidimensional Darboux sums and integrability are Lower and upper Darboux sums over a grid partition in , The lower and upper Darboux integrals over a nondegenerate rectangle in , and Riemann's criterion on a nondegenerate rectangle in : integrability is equivalent to arbitrarily small Darboux gaps.
A finite rectangle cover can be converted to grid cells meeting with arbitrarily small excess volume (A finite rectangle cover admits grid control with arbitrarily small volume excess), and rectangles are finite coordinate products (Axis-parallel rectangles in and their volume).
Proof
On a grid cell, the infimum of is exactly when the cell is contained in , while its supremum is exactly when the cell meets . Thus lower and upper sums are inscribed and covering grid approximations.
Apply [L3] to each finite outer rectangle approximation to obtain a grid whose cells meeting have arbitrarily small excess volume. For an inner approximation, shrink each nondegenerate inscribed rectangle by an arbitrarily small volume, insert the shrunken endpoints, and retain the grid cells inside it. Degenerate rectangles contribute zero. Splitting along the aligned endpoints shows that arbitrary Jordan approximations and grid approximations have the same infimum and supremum.
Equality of Jordan contents is therefore equality of the lower and upper integrals on the fixed bounding rectangle, and their common value is .
Depends on
- Jordan inner and outer content and Jordan measurable bounded sets in $\mathbb{R}^m$
- The lower and upper Darboux integrals over a nondegenerate rectangle in $\mathbb{R}^m$
- Lower and upper Darboux sums over a grid partition in $\mathbb{R}^m$
- Riemann's criterion on a nondegenerate rectangle in $\mathbb{R}^m$: integrability is equivalent to arbitrarily small Darboux gaps
- Grid partitions of a rectangle in $\mathbb{R}^m$, their cells, refinements and mesh
- Refinement raises multidimensional lower sums and lowers upper sums, with a quantitative boundary-slab estimate
- A finite rectangle cover admits grid control with arbitrarily small volume excess
- Axis-parallel rectangles in $\mathbb{R}^m$ and their volume
- Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space
Used by
- Cavalieri: Jordan content is the integral of sectional contents, and equal sections give equal content Corollary
- Jordan content is finitely additive when the overlap has content zero Corollary
- The content of a compact Jordan image is the integral of the absolute Jacobian determinant Corollary
- The divergence at a point is the limit of outward flux per unit volume Corollary
- The volume of a glued elementary solid is a third of the outward flux of the position field Corollary
- The rational points of [0,1]² form a bounded null set that is not Jordan measurable Counterexample
- The Riemann integral of a bounded function over a bounded Jordan measurable set Definition
- Both sides of the divergence theorem for F(x,y,z)=(x²,y²,z²) on the closed unit box Example
- The right triangle {(x,y)∈[0,1]²:x+y≤1} has Jordan content 1/2 Example
- Change of variables for a C¹ map injective and regular only on the interior of a compact Jordan set Lemma
- Finite Jordan covers bound upper integrals, while interior-disjoint Jordan subfamilies bound lower integrals Lemma
- Shared boundary arcs cancel when finitely many elementary regions are glued Lemma
- The plane Gaussian integral equals π by polar coordinates Lemma
- The Riemann integral over a Jordan set is independent of the bounding rectangle Lemma
- Riemann area between continuous graphs equals Jordan content Proposition
- A bounded set in ℝᵐ is Jordan measurable iff its boundary is null, equivalently of content zero Theorem
- Locally dominated parameter-dependent improper multiple integrals are continuous Theorem
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, Jordan Measurable Sets (standard reference, not scraped)
- J. Lebl, Basic Analysis, Outer Measure and Null Sets (standard reference, not scraped)