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The Saks–Henstock lemma for fine partial tagged partitions

Statement

If $f$ is Henstock–Kurzweil integrable on $[a, b]$ , then for every $ε > 0$ there is a gauge $δ$ such that every $δ$ -fine partial tagged partition ${([u_{i}, v_{i}], ξ_{i})}_{i = 1}^{m}$ satisfies

$\sum_{i = 1}^{m} ∣ f (ξ_{i}) (v_{i} - u_{i}) - \int_{u_{i}}^{v_{i}} f ∣ < ε .$

The assertion includes the empty partial partition.

Fine partial tagged partitions have uniformly small sums of local integration errors.

Facts & Assumptions

Given: An HK-integrable $f$ and a fine partial tagged partition for a sufficiently accurate gauge.

[L1]

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

[L2]

Henstock–Kurzweil integrals restrict to subintervals and add over adjacent intervals (Henstock–Kurzweil integrability on subintervals and additivity over adjacent intervals).

[L3]

HK integrability means that one gauge makes every fine tagged sum lie within a prescribed error of the integral value (The Henstock–Kurzweil integral on a compact interval).

Proof

technique · direct

1.1

The empty family has error $0$ . Otherwise fix a whole-interval gauge whose full-partition error is below $ε / 4$ . After a partial partition fine for that fixed gauge is given, [L2] makes $f$ integrable on each of its finitely many complementary compact intervals. For any prescribed complement error, [L3] supplies a local accuracy gauge there; [L1] supplies a partition fine for the minimum of that local gauge and the already fixed whole-interval gauge. Thus the resulting completions are both arbitrarily accurate and fine for the original gauge.

givenL1L2L3

2.1

For the cells with nonnegative local error, complete their complement with fine partitions whose total local error is below $ε / 4$ . Additivity [L2] identifies the resulting full-partition error with the selected positive errors plus those complement errors, so the positive total is below $ε / 2$ . Repeating the construction for the negative cells bounds the absolute value of their total by $ε / 2$ ; adding the two bounds gives the displayed strict estimate.

step 1.1L1L2algebra∎

Depends on

Used by

Hake's theorem: a finite-endpoint generalized integral is a proper Henstock–Kurzweil integral after assigning the endpoint value Theorem

Dependency tree · two levels

8 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)