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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
The lower and upper Darboux integrals over a nondegenerate rectangle in
Lower and upper Darboux sums over a grid partition in
Riemann's criterion on a nondegenerate rectangle in : integrability is equivalent to arbitrarily small Darboux gaps
Grid partitions of a rectangle in , 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 and their volume
Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space