The Riemann integral of a bounded function over a bounded Jordan measurable set

Depends on

• Jordan inner and outer content and Jordan measurable bounded sets in $\mathbb{R}^m$
• A bounded set is Jordan measurable iff its indicator is Riemann integrable, and the integral is its Jordan content
• The lower and upper Darboux integrals over a nondegenerate rectangle in $\mathbb{R}^m$
• 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
• Lower bound, bounded below, bounded set

Used by

• A curl-free field has zero circulation around the induced boundary chain of a C² patch Corollary
• A field with vanishing divergence has zero outward flux through the boundary of a glued elementary solid Corollary
• Area of an elementary Green region as a boundary line integral Corollary
• The divergence at a point is the limit of outward flux per unit volume Corollary
• The normal component of the curl is the limiting circulation per unit area of shrinking discs Corollary
• The volume of a glued elementary solid is a third of the outward flux of the position field Corollary
• A solid between continuous graphs over a compact Jordan base Definition
• Boundary presentations adapted to a simple solid region in a coordinate direction Definition
• Integral of a form over a smooth singular simplex Definition
• Regular parametrized surface patches on compact Jordan parameter regions Definition
• Sections, lower and upper section integrals, and iterated Riemann integrals on product rectangles and Jordan sets Definition
• Simple solid regions in a coordinate direction and their cyclic coordinate projection Definition
• A right circular cylinder is an elementary solid region, presented by two caps and four side quarters Example
• Both sides of the divergence theorem for F(x,y,z)=(x²,y²,z²) on the closed unit box Example
• The closed ball is an elementary solid region, presented by the eight spherical octants Example
• The closed unit box, with its six faces, is an elementary solid region Example
• The Mobius band presented by two regular patches, with normal comparison on the interiors of the overlap components Example
• A cyclic permutation of the coordinates of ℝ³ preserves Jordan measurability and integrals Lemma
• Additivity of the integral over finitely many Jordan pieces that fill a Jordan set up to content zero Lemma
• Change of variables for a C¹ map injective and regular only on the interior of a compact Jordan set Lemma
• Changing a bounded integrand on a content-zero set does not change its Riemann integral Lemma
• Finite Jordan covers bound upper integrals, while interior-disjoint Jordan subfamilies bound lower integrals Lemma
• Internal faces cancel and volume integrals add when elementary solid regions are glued Lemma
• Riemann–Lebesgue comparison for distribution test integrands Lemma
• Shared boundary arcs cancel when finitely many elementary regions are glued Lemma
• The flux of a single-component field through a graph face is a base integral of its trace Lemma
• The Riemann integral over a Jordan set is independent of the bounding rectangle Lemma
• The single-direction flux identity on a simple solid region Lemma
• Conventions and proved scope for the Riemann integral in ℝᵐ and Jordan content Remark
• A continuous real function on a compact Jordan measurable set is Riemann integrable over that set Theorem
• Absolute convergence makes signed improper multiple integrals independent of exhaustion Theorem
• Comparison and absolute comparison tests for improper multiple integrals Theorem
• Every Jordan exhaustion computes a nonnegative improper multiple integral Theorem
• Fubini over a bounded Jordan set when all but a content-zero family of sections are integrable Theorem
• The divergence theorem on an elementary solid region Theorem

Dependency tree · two levels

29 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)