TheoremStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-01

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Linearity, monotonicity, the absolute-value estimate and coordinate-slice additivity for the Riemann integral in $\mathbb{R}^m$

Statement

Let $Q=\prod_{j<m}[a_j,b_j]$ be nondegenerate. For integrable $f,g:Q\to\mathbb R$ and scalars $\alpha,\beta$ , the function $\alpha f+\beta g$ is integrable and its integral is $\alpha\int_Qf+\beta\int_Qg$ . If $f\le g$ , then $\int_Qf\le\int_Qg$ . Also $|f|$ is integrable and $|\int_Qf|\le\int_Q|f|$ . If $a_r<c<b_r$ , cutting $Q$ at the coordinate hyperplane $x_r=c$ gives two nondegenerate subrectangles; integrability on $Q$ is equivalent to integrability on both restrictions, and their integral values add to the integral over $Q$ .

Facts & Assumptions

Given: The stated integrable functions on the nondegenerate rectangle, and, for coordinate-slice additivity, a strictly interior cut $a_r<c<b_r$ .

[L1]

Small Darboux gaps characterize integrability; a common refinement improves both lower and upper sums; and the common integral is the tagged-mesh limit (Riemann's criterion on a nondegenerate rectangle in $\mathbb{R}^m$ : integrability is equivalent to arbitrarily small Darboux gaps, Refinement raises multidimensional lower sums and lowers upper sums, with a quantitative boundary-slab estimate, The multidimensional Darboux and tagged-mesh definitions of the Riemann integral agree).

[L2]

Grid sums split coordinatewise (Grid partitions of a rectangle in $\mathbb{R}^m$ , their cells, refinements and mesh, Laws of finite sums and finite products).

[L3]

$\bigl||u|-|v|\bigr|\le|u-v|$ , the reverse triangle inequality on the real line (The reverse triangle inequality, Absolute value in an ordered field, Basic properties of the absolute value).

Proof

technique · direct

1.1

Refine grids good for $f$ and $g$ . Cellwise supremum and infimum estimates make the gap of $\alpha f+\beta g$ at most $|\alpha|$ times the gap of $f$ plus $|\beta|$ times that of $g$ ; tagged-sum linearity identifies the value.

L1L2

1.2

Termwise $f\le g$ gives monotonicity of every tagged sum and hence of integrals. By [L3], the oscillation of $|f|$ on a cell is no larger than that of $f$ , so $|f|$ is integrable; $-|f|\le f\le|f|$ then gives the absolute-value estimate.

L1L3given

1.3

Insert the cut coordinate into the grid. [L2] splits every Darboux or tagged sum into the two subrectangle sums. Good grids splice conversely, proving integrability on $Q$ exactly when both restrictions are integrable, and proving additivity.

L1L2

2.1

These arguments establish all clauses with positively oriented rectangles.

step 1.1step 1.2step 1.3∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 73 results over 20 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

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)

Linearity, monotonicity, the absolute-value estimate and coordinate-slice additivity for the Riemann integral in Rm\mathbb{R}^mRm