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Henstock–Kurzweil integrability on subintervals and additivity over adjacent intervals

Statement

Let $a \leq c \leq b$ . A function is Henstock–Kurzweil integrable on $[a, b]$ if and only if its restrictions to $[a, c]$ and $[c, b]$ are integrable, and then

$\int_{a}^{b} f = \int_{a}^{c} f + \int_{c}^{b} f .$

For compact HK integrals, define the oriented value by $\int_{v}^{u} f : = - \int_{u}^{v} f$ when $u < v$ and $\int_{u}^{u} f : = 0$ . With this convention, for points $u, v, w$ , $\int_{u}^{w} f = \int_{u}^{v} f + \int_{v}^{w} f$ whenever the compact pieces are integrable.

For points $u, v, w$ , $\int_{u}^{w} f = \int_{u}^{v} f + \int_{v}^{w} f$ whenever the compact pieces are integrable.

Henstock–Kurzweil integrals restrict to subintervals and add over adjacent intervals.

Facts & Assumptions

Given: A function on $[a, b]$ and a cut point $c \in [a, b]$ .

[L1]

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

[L2]

A function on a compact interval is Henstock–Kurzweil integrable if and only if, for every $ε > 0$ , there is a gauge such that every pair of fine tagged sums differs by less than $ε$ (The Cauchy criterion for Henstock–Kurzweil integrability).

Proof

technique · direct

1.1

For the forward direction, fix a whole-interval gauge whose fine sums are within $ε / 2$ of the integral. Given two fine partitions $P, Q$ of $[a, c]$ , use [L1] to choose one fine partition $R$ of $[c, b]$ for the restricted gauge. Then $P \cup R$ and $Q \cup R$ are whole-interval fine partitions, so $∣ S (f, P) - S (f, Q) ∣ < ε$ ; [L2] proves integrability on $[a, c]$ , and the symmetric completion proves it on $[c, b]$ .

givenL1L2algebra

1.2

For the reverse direction, choose side gauges for error $ε / 2$ . For $x < c$ shrink the left gauge below $(c - x) / 2$ , for $x > c$ shrink the right gauge below $(x - c) / 2$ , and at $c$ take the minimum of the two gauges. Thus a fine cell can cross $c$ only when tagged at $c$ , in which case splitting it at $c$ produces one fine cell for each side. The two side estimates then add, proving whole-interval integrability and the displayed additivity, including $c = a$ or $c = b$ .

givenalgebra

2.1

Order $u, v, w$ , apply step 1.2 on the two adjacent compact subintervals, and reverse any necessary limits with the orientation convention in the Statement. The resulting signed equality is the oriented three-point identity in every ordering.

step 1.2algebra∎

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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)