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Every continuous function on a closed nondegenerate rectangle in is Riemann integrable
Statement
Every continuous real function on a closed nondegenerate rectangle , , is Riemann integrable.
Facts & Assumptions
Given: A continuous .
is compact by Heine-Borel (Heine-Borel in : with the Euclidean metric a subset of is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line, Open cover, subcover, compact metric space, and compact subset of a metric space).
A continuous function on a compact metric space is uniformly continuous and bounded (Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous, The image of a compact metric space under a continuous map is compact, and so is the image of any compact subset, A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value, Continuity of a map between metric spaces, at a point and globally, in the - form).
The Euclidean and sup-norm metrics are the published metrics and satisfy (The finite and reverse triangle inequalities for a norm; and for every norm on satisfies and is Lipschitz, hence continuous, for clause 3, Each is a norm on , and the induced metrics are exactly , and of the published metric-spaces page).
Arbitrarily small Darboux gaps characterize integrability (Riemann's criterion on a nondegenerate rectangle in : integrability is equivalent to arbitrarily small Darboux gaps).
Proof
Given , use [L2] with oscillation target and choose a grid whose mesh is below the resulting sup-metric radius.
Every cell then has oscillation below that target. Since cell volumes sum to , the Darboux gap is below .
The multidimensional Riemann criterion proves integrability.
Depends on
- Riemann's criterion on a nondegenerate rectangle in $\mathbb{R}^m$: integrability is equivalent to arbitrarily small Darboux gaps
- Lower and upper Darboux sums over a grid partition in $\mathbb{R}^m$
- Grid partitions of a rectangle in $\mathbb{R}^m$, their cells, refinements and mesh
- Axis-parallel rectangles in $\mathbb{R}^m$ and their volume
- Heine-Borel in $\mathbb{R}^n$: with the Euclidean metric a subset of $\mathbb{R}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line
- Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous
- Continuity of a map between metric spaces, at a point and globally, in the $\varepsilon$-$\delta$ form
- Open cover, subcover, compact metric space, and compact subset of a metric space
- The finite and reverse triangle inequalities for a norm; and for $n \ge 1$ every norm $N$ on $\mathbb{R}^n$ satisfies $N(x) \le C\lVert x\rVert_1$ and is Lipschitz, hence continuous, for $d_2$
- Each $\lVert\cdot\rVert_p$ is a norm on $\mathbb{R}^n$, and the induced metrics are exactly $d_1$, $d_2$ and $d_\infty$ of the published metric-spaces page
- The image of a compact metric space under a continuous map is compact, and so is the image of any compact subset
- A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value
- Lower bound, bounded below, bounded set
Used by
- A C¹ diffeomorphism satisfies the change-of-variables formula for L¹ functions Corollary
- A continuous function on a closed rectangle has repeated Riemann integrals in every coordinate order, all equal to its multiple integral Corollary
- The integral of a product function on a product rectangle is the product of the two integrals Corollary
- A deterministic integral construction of a Gaussian process Example
- A normalized compactly supported top form on Euclidean space Example
- Principal value distribution one over x Example
- Borel change of variables from the compact-support formula and Radon uniqueness Lemma
- Finite chart localization gives choice-free integration and compact Stokes Lemma
- Zero-integral compactly supported top forms on Euclidean space have compactly supported primitives Lemma
- A jointly continuous finite-interval parameter integral of holomorphic functions is holomorphic Theorem
Dependency tree · two levels
84 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)