TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21

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Monotonicity of the Henstock–Kurzweil integral

Statement

If $f$ and $g$ are Henstock–Kurzweil integrable on $[a, b]$ and $f (x) \leq g (x)$ throughout the interval, then

$\int_{a}^{b} f \leq \int_{a}^{b} g .$

In particular, $m \leq f \leq M$ implies $m (b - a) \leq \int_{a}^{b} f \leq M (b - a)$ .

Facts & Assumptions

Given: HK-integrable $f, g$ on $[a, b]$ with $f \leq g$ .

[L1]

The Henstock–Kurzweil integral is linear (Linearity of the Henstock–Kurzweil integral).

[L2]

Every gauge admits a fine tagged partition (Cousin's lemma: every gauge on a compact interval admits a fine tagged partition).

Proof

technique · contradiction

1.1givenL2assume-contra

If a nonnegative integrable function $h$ had integral $H < 0$ , choose a gauge making every fine sum differ from $H$ by less than $- H / 2$ and use [L2]; every such sum is nonnegative, contradicting $S (h, P) < H / 2 < 0$ .

2.1step 1.1L1algebradischarge-contradiction∎

Apply step 1.1 to $h = g - f$ and use [L1] to obtain $0 \leq \int (g - f) = \int g - \int f$ . Every constant $k$ is HK integrable with integral $k (b - a)$ because every tagged sum equals that value; applying the first conclusion to $f - m$ and $M - f$ therefore gives the constant bounds, with equality on a degenerate interval.

Depends on

Used by

Dependency tree · two levels

19 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

Alessandro Fonda, The Kurzweil-Henstock Integral for Undergraduates, Ch. 1 (standard reference, not scraped)
Andrew Bruckner, Judith Bruckner and Brian Thomson, Real Analysis, Sections 1.2 and 1.21 (standard reference, not scraped)