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Let $[a,b]$ be an interval in $\mathbb{R}$. Let $P$ be a finite partition of $[a,b]$.

Define $T_n=\{a+\frac{b-a}{n}i \in \mathbb{R}| 0≦i≦n\}$ for each $n\in\mathbb{Z}^+$. (Let's denote $T_n=\{t_0,...,t_n\}$ for convenience)

My questions is, does there exists a finite refinement $Q=\{x_0,...,x_m\}$ of $P$ and $T_N$ satisfy below two properties?

That is;

(i) $i≠j\bigwedge t_i\in [x_k,x_{k+1}]\bigwedge t_j\in [x_l,x_{l+1}] \Rightarrow k≠l$

(ii) $\forall i≦N, \exists k such that $t_i\in[x_k,x_{k+1}]$

(iii) $\forall k such that $t_i\in[x_k,x_{k+1}]$

I believe this is false, but i cannot figure out how to show that..

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    Wasn't this question asked earlier: http://math.stackexchange.com/questions/267839/if-f-is-riemann-stieltjes-integrable-then-does-there-exist-a-partition-of-whi/267954#2679542012-12-31
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    @emka Yes, but i posted it again (with more clarity) because i didn't mention the definition of Riemann-Stieltjes Integral and I looked up 'Zygmund&Wheeden's text' in leo's answer and i found that $f$ may be integrable with reapect to the definition of the text, but may not be integrable with respect to the definition of mine2012-12-31
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    @Christian How do you prove that your partition $Q$ satisfies properties (i)&(ii)?2013-01-01
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    @Christian Blatter: That's the first thing that came to my mind also. Katlus, Why doesn't Christian's solution work?2013-01-01
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    I have missed one more property in my mind.. Now it's edited.2013-01-01

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