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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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    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