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The indicator of the irrationals is Henstock–Kurzweil integrable with integral
Example
Let be the indicator of the irrationals. The indicator of the irrationals is Henstock–Kurzweil integrable with integral , but it is not Riemann integrable.
The indicator of the irrationals is Henstock–Kurzweil integrable with integral and is not Riemann integrable.
The indicator of the irrationals is Henstock–Kurzweil integrable with integral .
Facts & Assumptions
Given: The function for irrational and for rational .
The rationals are countably infinite: ( is countably infinite).
The rationals and the irrationals are both dense in (Both and are dense in , and every nonempty open subset of is uncountable).
Darboux integrability means equality of the lower and upper Darboux integrals (The lower and upper Darboux integrals of a bounded on as and , Darboux integrability as their equality, and the notation ).
HK integrability requires one gauge to control every fine tagged Riemann sum (The Henstock–Kurzweil integral on a compact interval).
Verification
Enumerate the rationals in as using [L1]; for a requested , choose the gauge at below and choose any fixed positive gauge at irrational tags.
In a fine partition, a fixed tag occurs on at most two cells, so the total length of rational-tagged cells is below ; hence , and [L4] gives the stated HK value.
By [L2], every partition cell contains points where and points where , so every lower Darboux sum is and every upper Darboux sum is ; [L3] therefore rules out Riemann integrability.
Depends on
- The Henstock–Kurzweil integral on a compact interval
- Cousin's lemma: every gauge on a compact interval admits a fine tagged partition
- $\mathbb{Q}$ is countably infinite
- The Dirichlet function $1_{\mathbb{Q}}$, and Thomae's function $t$ with $t(x) = 1/q$ at a rational $x = p/q$ in lowest terms with $q \ge 1$ and $t(x) = 0$ at every irrational $x$
- Both $\mathbb{Q}$ and $\mathbb{R} \setminus \mathbb{Q}$ are dense in $\mathbb{R}$, and every nonempty open subset of $\mathbb{R}$ is uncountable
- Partition of $[a,b]$ as a finite strictly increasing list $a = t_0 < t_1 < \dots < t_n = b$, its subintervals and their lengths, its mesh, refinement, and the common refinement of two partitions
- For bounded $f$ on $[a,b]$ and a partition $P$: the infimum $m_i$ and supremum $M_i$ of $f$ on the $i$-th subinterval, and the lower and upper Darboux sums $L(f,P) = \sum_i m_i \Delta_i$ and $U(f,P) = \sum_i M_i \Delta_i$
- The lower and upper Darboux integrals of a bounded $f$ on $[a,b]$ as $\sup_P L(f,P)$ and $\inf_P U(f,P)$, Darboux integrability as their equality, and the notation $\int_a^b f$
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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, Exercise 1:21.2 (standard reference, not scraped)