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Let $\alpha$ be a monotonically increasing function.

Say, $f\in\mathscr{R}(\alpha)$.

Then does there exist a partition $P=\{x_0,...,x_n\}$ such that $$x_i=a+ \frac{b-a}{n}i,$$ $i\in\{0,\ldots,n\}$ and $$U(P,f,\alpha)-L(P,f,\alpha)<\epsilon$$ for each $\epsilon>0$?

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Is this a proof of the right-hand rule introduced in early in Calc II? –  emka Dec 31 '12 at 0:26
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1 Answer 1

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This Theorem is from the book Measure and Integral by Zygmund & Wheeden:

Theorem (2.29) Zygmund

Proof1

Proof2

According to this given $\epsilon\gt 0$ there exist a $\delta\gt 0$ such that for any partition $\Gamma$, if $|\Gamma|\lt\delta$, then $$U_\Gamma-L_\Gamma\lt\epsilon.$$

So, if your $f$ is bounded (it must be, otherwise the $U(P,f,\alpha)$ or $L(P,f,\alpha)$ might have no sense), given $\epsilon\gt 0$, in order to pick a uniform partition $$P=\{a=x_0\lt\cdots\lt x_n=b\}$$ such that $$U(P,f,\alpha)-L(P,f,\alpha)\lt\epsilon,$$ it is enough to choose $n$ large enough so that $$\frac{b-a}{n}\lt\delta.$$

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would you please tell me how to prove above theorem? –  Katlus Dec 31 '12 at 6:25
    
@Katlus I've added the proof from the book. I recommend you find the book at your library and read the chapter (1 and) 2. –  leo Dec 31 '12 at 21:09
    
Thanks for the posting, but there is a problem. Note that the definition of riemann-Stieltjes integral by Zygmund&Wheeden is 'different' from the definition using upper-sum and lower-sum. My definition for Riemann-Stieltjes integral is by using Upper-sum and Lower-sum so the proof above fails with respect to my definition –  Katlus Jan 1 '13 at 12:34
    
Let $\epsilon>0$ be given. Note that $\inf_{P} U(P,f,\alpha)$ may differ from $\inf_{n\in\mathbb{N}} U(T_n,f,\alpha)$ where $T_n=\{a+\frac{b-a}{n}i\in [a,b]| 0≦i≦n\}$. Do you see why above proof fails with respect to my definition? One should prove first that these two infima the same and this is actually what I'm asking. –  Katlus Jan 1 '13 at 12:39
    
I read Zygmund&Wheeden's book yesterday in the library, and you can see that he even said 'There are functions which are not ingerable w.r.t the definition in the book, but are integrable w.r.t the definition by Upper-Sum and Lower-Sum. –  Katlus Jan 1 '13 at 12:45
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