TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21

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Linearity of the Henstock–Kurzweil integral

Statement

The Henstock–Kurzweil integral is linear. If $f, g$ are integrable on $[a, b]$ and $c, d \in R$ , then $c f + d g$ is integrable and

$\int_{a}^{b} (c f + d g) = c \int_{a}^{b} f + d \int_{a}^{b} g .$

Facts & Assumptions

Given: HK-integrable functions $f, g$ and scalars $c, d$ .

[L1]

For every $ε > 0$ , one gauge controls every fine tagged Riemann sum of an HK-integrable function (The Henstock–Kurzweil integral on a compact interval).

[L2]

Finite sums are additive and commute with scalar multiplication (Laws of finite sums and finite products).

Proof

technique · direct

1.1

For $f + g$ , take the pointwise minimum of gauges from [L1] with half the requested error; by [L2], $S (f + g, P) = S (f, P) + S (g, P)$ , and the triangle inequality gives the required estimate.

givenL1L2

2.1

For a scalar multiple, the case of scalar $0$ is immediate, and otherwise [L1] with tolerance $ε / ∣ c ∣$ and [L2] gives $\int c f = c \int f$ ; combining the sum and scaling conclusions proves the formula.

step 1.1L1L2algebra∎

Depends on

Used by

Dependency tree · two levels

18 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)