ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passaudited 2026-08-01

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The unit box in $R^{m}$ has volume $1$ , and the integral of a constant $c$ over it is $c$

Example

For $Q = [0, 1]^{m}$ , $vol (Q) = 1$ , and for every constant $c$ , $\int_{Q} c = c$ .

Facts & Assumptions

Given: $m \geq 1$ and constant $c$ .

[L1]

Rectangle volume is the finite product of side lengths (Axis-parallel rectangles in $R^{m}$ and their volume).

[L2]

Darboux sums and the integral are Lower and upper Darboux sums over a grid partition in $R^{m}$ and The lower and upper Darboux integrals over a nondegenerate rectangle in $R^{m}$ .

Verification

technique · direct

1.1

Every side has length $1$ , so the finite product volume is $1$ .

L1given

1.2

On every cell, both infimum and supremum of the constant function are $c$ . Thus both sums are $c \sum_{i} vol (Q_{i}) = c$ .

L2given

2.1

Lower and upper integrals therefore both equal $c$ .

step 1.2L2∎

Depends on

Used by

The rational points of [0,1]² form a bounded null set that is not Jordan measurable Counterexample

Dependency tree · two levels

32 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)
J. Lebl, Basic Analysis, Riemann Integral in Several Variables (standard reference, not scraped)

The unit box in RmRm has volume 11, and the integral of a constant cc over it is cc