Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
The Henstock–Kurzweil integral on a compact interval
Definition
Let and . The function is Henstock–Kurzweil integrable on with value when, for every , there is a gauge on such that
for every -fine tagged partition . Thus, for every one gauge controls every fine tagged Riemann sum. Cousin's lemma ensures that the quantified class of fine partitions is nonempty.
For every one gauge controls every fine tagged Riemann sum.
The value, once uniqueness is proved, is written . On a degenerate interval, the Henstock–Kurzweil integral is .
Depends on
Used by
- The indefinite Henstock–Kurzweil integral of a derivative is a primitive Corollary
- Henstock–Kurzweil integrals on half-open and unbounded intervals by compact truncation limits Definition
- 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
- Every Riemann integrable function is Henstock–Kurzweil integrable with the same integral 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
14 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)