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Every Riemann integrable function is Henstock–Kurzweil integrable with the same integral
Statement
Every Riemann integrable function is Henstock–Kurzweil integrable with the same integral.
Facts & Assumptions
Given: A Riemann integrable on with value .
For every , Riemann integrability gives such that every tagged partition of mesh below has sum within of (The Darboux and Riemann definitions agree: a bounded on is Darboux integrable with integral if and only if for every real there is a real such that for every tagged partition of mesh below ).
A tagged partition is gauge-fine when every cell lies in its tag's centered gauge interval (Gauges and gauge-fine tagged partitions of a compact interval).
Proof
If , both integrals are ; otherwise take the constant gauge from [L1], so [L2] makes every -fine cell shorter than and the whole partition has mesh below .
The universal estimate in [L1] therefore applies to every -fine tagged partition, which is exactly the HK definition with the same value .
Depends on
- The Henstock–Kurzweil integral on a compact interval
- The Darboux and Riemann definitions agree: a bounded $f$ on $[a,b]$ is Darboux integrable with integral $I$ if and only if for every real $\varepsilon > 0$ there is a real $\delta > 0$ such that $|S(f,P,\xi) - I| < \varepsilon$ for every tagged partition of mesh below $\delta$
- Tagged partitions of $[a,b]$, with a tag $\xi_i$ in each subinterval, and the Riemann sum $S(f,P,\xi) = \sum_i f(\xi_i)\,\Delta_i$
- Gauges and gauge-fine tagged partitions of a compact interval
Used by
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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)