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a) Please help in proving that if $f$ is Riemann integrable on $[a,b]$ for some $a \lt b$, then $f$ is Henstock-Kurzweil integrable on $[a,b]$ with Henstock-Kurzweil integral equal to the Riemann integral of $f$, from $a$ to $b$.

b) Please show that $1_{\mathbb{Q} \cap [a,b]}$ is Henstock-Kurzweil integrable on $[a,b]$ for all $a \lt b$.

Can someone please explain the problem? I'm not understanding the hint.

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By "1 subscript rations" you mean $1_{\mathbb Q}$, i.e. the characteristic function of rationals? In the part a), in which sense is the function f integrable? Lebesgue? Riemann? –  Martin Sleziak Apr 16 '11 at 8:19
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I tried to fix the TeX and tried to clarify your question. I hope I have captured the intention. Two more things: Please show your own thoughts on this (since you tagged this as homework) and please start accepting answers by clicking on the gray checkmark sign on the left. –  t.b. Apr 16 '11 at 8:36
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@user8917: what textbook are you using that covers Henstock-Kurwzweil integration and does not include proofs of these basic facts? –  Pete L. Clark Apr 16 '11 at 8:43
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See Examples 3.1 and 3.2 in this text mathdl.maa.org/images/upload_library/22/Ford/Bartle625-632.pdf Robert G. Bartle, Return to the Riemann integral, American Mathematical Monthly, vol. 103 (1996), no. 8, 625--632 –  Martin Sleziak Apr 16 '11 at 12:48
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Well, the first example answers your first question and the second one your second. –  t.b. Apr 16 '11 at 23:32

1 Answer 1

Hint for (b). Let $\epsilon>0$ and define a gauge $\Delta$ on $[a,b]$ by taking $\Delta(r_k) = \epsilon/2^k$ where $r_k$ is the $k$th rational and $\Delta(x)=1$ for irrational $x$. Then show any $\Delta$-fine Riemann sum is small.

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