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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?2012-12-31

1 Answers 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?2012-12-31
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    @Katlus I've added the proof from the book. I recommend you find the book at your library and read the chapter (1 and) 2.2012-12-31
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    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 definition2013-01-01
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    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.2013-01-01
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    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.2013-01-01
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    @Katlus I don't know what is your definition using upper and lower sums. Would you please add it in your post? Could you say what book or notes (if possible) are you following? There are several, not necessarily equivalent, ways to introduce Riemann-Stieltjes integral.2013-01-01
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    I'm following the definition in Rudin PMA. There are differences between Rudin's and Zygmund's. For example, Zygmund's definition does not work when $f,\alpha$ share discontinuities, but Rudin's definition does work when $f,\alpha$ does not share discontinuities at the same side. That is, if is not continuous from the right and continuous from the left, and $\alpha$ is not continuous from the left and continuous from the right, then it is Riemann-Stieltjes Integrable with respect to Rudin's Definition.2013-01-02
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    @Katlus I see. You can edit your question to include those details, in order to get good answers. I'll delete this answer.2013-01-02
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    No please leave this.. I didn't know that there are different definitions before i post this question, so your post helped me. Thank you2013-01-03