LemmaStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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The Riemann integral of a compactly supported function is independent of its bounding rectangle

Statement

Let $n\ge1$ and let $f:\mathbb R^n\to\mathbb R$ have compact support. If $f$ is Riemann integrable on one closed rectangle whose interior contains its support, then it is integrable on every such rectangle, and all the resulting integrals are equal. This includes the empty-support case.

Facts & Assumptions

Given: Compactly supported $f$ and bounding rectangles $Q_1,Q_2$ whose interiors contain its support.

[L1]

Extending an integrable function on a Jordan set by zero to a bounding rectangle gives a well-defined integral independent of that rectangle (The Riemann integral over a Jordan set is independent of the bounding rectangle).

[L2]

Cutting rectangles along coordinate hyperplanes preserves integrability and adds the integrals of the pieces (Linearity, monotonicity, the absolute-value estimate and coordinate-slice additivity for the Riemann integral in $\mathbb{R}^m$ ).

Proof

technique · common-extension

1.1

Choose a third rectangle $Q$ whose interior contains $Q_1\cup Q_2$ . Since $f=0$ outside its support, extending $f|_{Q_i}$ by zero to $Q$ recovers exactly $f|_Q$ .

given

2.1

If $f|_{Q_1}$ is integrable, [L1] makes its zero extension integrable on $Q$ with the same integral. Restricting this function to $Q_2$ by the coordinate cuts in [L2] gives integrability there, again with zero contribution off the support.

L1L2step 1.1

3.1

Applying [L1] to $Q_1$ and $Q_2$ inside the common rectangle yields $\int_{Q_1}f=\int_Qf=\int_{Q_2}f$ . If the support is empty, all three functions are identically zero, so the same argument gives value $0$ .

L1step 2.1∎

Depends on

Used by

A compactly supported Riemann integrand admits the global change-of-variables formula from a diffeomorphism near the relevant compact preimage Corollary

Cited to discharge well-definedness by The support of a function on ℝⁿ and its compactly supported Riemann integral.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 85 results over 22 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

A. Leibman, Multidimensional Real Analysis, §5.5 (standard reference, not scraped)