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The Cauchy criterion for Henstock–Kurzweil integrability

Statement

A function $f : [a, b] \to R$ is Henstock–Kurzweil integrable if and only if for every $ε > 0$ there is a gauge $δ$ such that

$∣ S (f, P) - S (f, Q) ∣ < ε$

for every pair of $δ$ -fine tagged partitions $P, Q$ .

Facts & Assumptions

Given: A function $f$ on a compact interval.

[L1]

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

[L2]

A nested sequence of closed intervals whose lengths tend to zero has a one-point intersection (A nested sequence of nonempty closed bounded intervals has nonempty intersection, and the intersection is a single point exactly when the lengths tend to $0$ ).

[L3]

Countable choice provides a function selecting one member from each nonempty set in a family indexed by $N$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[L4]

If $∣ r ∣ < 1$ , then $r^{k} \to 0$ (For $∣ r ∣ < 1$ the sequence $r^{k}$ is null, and for $∣ r ∣ > 1$ the sequence $∣ r ∣^{k}$ diverges to $+ \infty$ ).

[L5]

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

Proof

technique · direct

1.1

For the forward direction, apply [L5] with error $ε / 2$ ; any two sums fine for the resulting gauge differ by less than $ε$ .

givenL5algebra

1.2

For the reverse direction, use [L3] to choose a diameter-controlling gauge $γ_{n}$ for tolerance $2^{- n}$ , set $δ_{n} = \min_{k \leq n} γ_{k}$ , and let $H_{n}$ be the closed interval hull of the nonempty set of $δ_{n}$ -fine sums supplied by [L1]; the $H_{n}$ are nested, have length at most $2^{- n}$ , and [L4] and [L2] give a common point $I$ .

givenL1L2L3L4

2.1

Given $ε > 0$ , [L4] gives $n$ with $2^{- n} < ε$ ; every $δ_{n}$ -fine sum lies in $H_{n}$ with $I$ , hence within $ε$ of $I$ , which is precisely HK integrability and selects no partition.

step 1.2L4algebra∎

Depends on

Used by

Henstock–Kurzweil integrability on subintervals and additivity over adjacent intervals Theorem

Dependency tree · two levels

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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)