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Cousin's lemma: every gauge on a compact interval admits a fine tagged partition
Statement
For , every gauge on a compact interval admits a fine tagged partition.
Equivalently, every gauge admits a fine tagged partition, and every gauge admits at least one fine tagged partition. In particular, every gauge on each complementary compact interval admits a fine tagged partition.
Facts & Assumptions
Given: A gauge on .
A nested sequence of nonempty closed bounded intervals whose lengths tend to zero has an intersection consisting of a single point (A nested sequence of nonempty closed bounded intervals has nonempty intersection, and the intersection is a single point exactly when the lengths tend to ).
A tagged partition is fine when every tagged cell lies inside its tag's gauge interval (Gauges and gauge-fine tagged partitions of a compact interval).
Proof
If , the declared degenerate partition is fine; otherwise suppose, for contradiction, that has no fine partition, bisect it, and at each stage retain the left half if it has no fine partition and otherwise the right half, which must have none because two fine half-partitions concatenate; the retained closed intervals are nested and have length , so [L2] and [L1] give one common point .
Since and the retained lengths tend to zero, a sufficiently late retained interval lies inside ; tagged by , [L3] makes that one cell a fine partition, contradicting its construction.
Depends on
- Gauges and gauge-fine tagged partitions of a compact interval
- A nested sequence of nonempty closed bounded intervals has nonempty intersection, and the intersection is a single point exactly when the lengths tend to $0$
- For $|r| < 1$ the sequence $r^k$ is null, and for $|r| > 1$ the sequence $|r|^k$ diverges to $+\infty$
Used by
- The Henstock–Kurzweil integral on a compact interval Definition
- Cousin's lemma yields the Heine–Borel theorem on a compact interval Example
- The indicator of the irrationals is Henstock–Kurzweil integrable with integral 1 Example
- The Henstock–Kurzweil integral has at most one value Proposition
- Every derivative is Henstock–Kurzweil integrable and satisfies Newton–Leibniz Theorem
- Henstock–Kurzweil integrability on subintervals and additivity over adjacent intervals Theorem
- Linearity of the Henstock–Kurzweil integral Theorem
- Monotonicity of the Henstock–Kurzweil integral Theorem
- The Cauchy criterion for Henstock–Kurzweil integrability Theorem
- The Saks–Henstock lemma for fine partial tagged partitions Theorem
Dependency tree · two levels
33 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)