TheoremStatement: AI-adaptedProof: AI-adaptedprecheck passaudited 2026-08-01

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

Statement

Let $Q = \prod_{j < m} [a_{j}, b_{j}]$ be nondegenerate. For integrable $f, g : Q \to R$ and scalars $α, β$ , the function $α f + β g$ is integrable and its integral is $α \int_{Q} f + β \int_{Q} g$ . If $f \leq g$ , then $\int_{Q} f \leq \int_{Q} g$ . Also $∣ f ∣$ is integrable and $∣ \int_{Q} f ∣ \leq \int_{Q} ∣ f ∣$ . 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 $R^{m}$ : 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 $R^{m}$ , their cells, refinements and mesh, Laws of finite sums and finite products).

[L3]

$∣ ∣ u ∣ - ∣ v ∣ ∣ \leq ∣ u - v ∣$ , 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

1.1L1L2

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

1.2L1L3given

Termwise $f \leq g$ 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; $- ∣ f ∣ \leq f \leq ∣ f ∣$ then gives the absolute-value estimate.

1.3L1L2

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.

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

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

Linearity, monotonicity, the absolute-value estimate and coordinate-slice additivity for the Riemann integral in RmRm

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

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