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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

1.1

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.

givenL1L2

2.1

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

step 1.1L1L2∎

Depends on

Used by

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

Dependency tree · two levels

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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