CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-01

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

At $m = 1$ , nondegenerate multidimensional rectangles, grid sums and the integral are exactly the published one-dimensional notions

Statement

Under $R^{1} ≅ R$ , nondegenerate multidimensional rectangles, grids, Darboux sums, tagged sums, integrability, and integral values are exactly the published one-dimensional notions on intervals $[a, b]$ with $a < b$ .

Facts & Assumptions

Given: The standard isometry $R^{1} ≅ R$ ( $R^{n}$ as the set of functions $n \to R$ , and $d_{1}$ , $d_{2}$ , $d_{\infty}$ are metrics on it, Each $∥ \cdot ∥_{p}$ is a norm on $R^{n}$ , and the induced metrics are exactly $d_{1}$ , $d_{2}$ and $d_{\infty}$ of the published metric-spaces page).

[L1]

The one-dimensional notions are The lower and upper Darboux integrals of a bounded $f$ on $[a, b]$ as $\sup_{P} L (f, P)$ and $\inf_{P} U (f, P)$ , Darboux integrability as their equality, and the notation $\int_{a}^{b} f$ , Tagged partitions of $[a, b]$ , with a tag $ξ_{i}$ in each subinterval, and the Riemann sum $S (f, P, ξ) = \sum_{i} f (ξ_{i}) Δ_{i}$ , and The Darboux and Riemann definitions agree: a bounded $f$ on $[a, b]$ is Darboux integrable with integral $I$ if and only if for every real $ε > 0$ there is a real $δ > 0$ such that $∣ S (f, P, ξ) - I ∣ < ε$ for every tagged partition of mesh below $δ$ .

[L2]

A multidimensional rectangle is a finite coordinate product with product volume; a grid is a coordinatewise partition whose cells split that volume; and the Darboux and tagged notions are the cited cell sums and their extrema or mesh limits (Axis-parallel rectangles in $R^{m}$ and their volume, Grid partitions of a rectangle in $R^{m}$ , their cells, refinements and mesh, Lower and upper Darboux sums over a grid partition in $R^{m}$ , The lower and upper Darboux integrals over a nondegenerate rectangle in $R^{m}$ , Tagged grid partitions and Riemann sums in $R^{m}$ , The multidimensional Darboux and tagged-mesh definitions of the Riemann integral agree).

Proof

technique · direct

1.1givenL1L2

With one coordinate, nondegeneracy says $a < b$ . A grid is one ordinary partition of $[a, b]$ , its cells are its subintervals, and their volumes are their lengths. The iterated cell sum has one index and is the ordinary finite sum.

2.1step 1.1L1L2∎

Therefore the lower, upper, and tagged sums agree term for term; taking extrema or mesh limits gives identical integrability classes and values.

Depends on

Used by

Conventions and proved scope for the Riemann integral in ℝᵐ and Jordan content Remark

Dependency tree · two levels

63 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)

At m=1m=1, nondegenerate multidimensional rectangles, grid sums and the integral are exactly the published one-dimensional notions