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

(iii) $\forall k<m, \exists i≦N$ 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: math.stackexchange.com/questions/267839/… –  emka Dec 31 '12 at 7:18
    
@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 mine –  Katlus Dec 31 '12 at 7:24
    
@Christian How do you prove that your partition $Q$ satisfies properties (i)&(ii)? –  Katlus Jan 1 '13 at 12:29
    
@Christian Blatter: That's the first thing that came to my mind also. Katlus, Why doesn't Christian's solution work? –  William Jan 1 '13 at 13:07
    
I have missed one more property in my mind.. Now it's edited. –  Katlus Jan 1 '13 at 13:37
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1 Answer

Hint: There are such sets $Q$. Think about what each of the conditions mean intuitively.

i) No two of the $t_i$ are in the same subinterval of $Q$.

ii) Each $t_i$ is in some subinterval of $Q$ (This seems a slightly unusual conition, surely any partition of $[a,b]$ will satisfy this property for any subset of $[a,b]$?).

iii) Each subinterval of $Q$ has some element of $t_i$ in it.

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Would you give me some direct hint? By the way i know what exactly coditions mean since i formulated it. I just wanted to create a refinement $Q$ of a given partition such that each length of subintervals in $Q$ is the same and each element of a partition $P$ is in the subinterval in $Q$. –  Katlus Jan 2 '13 at 20:03
    
More precisely, given $\epsilon>0$ and a real function $f$, suppose there exists a partition $P$ of $[a,b]$ such that $U(P,f,\alpha)-L(P,f,\alpha)< \epsilon$. I want to know if there exists a refinement $Q$ of $P$ such that each length of subintervals in $Q$ is the same and $U(Q,f,\alpha)-L(Q,f,\alpha)<\epsilon$. –  Katlus Jan 2 '13 at 20:06
    
@Katlus I'm not exactly sure what you mean by your comments, but I think that the partition $\{t_0, \frac12(t_0 + t_1), t_1, \frac12(t_1+t_2),\dots, t_n\}$ satisfies all three of the properties in the question. –  Tom Oldfield Jan 2 '13 at 22:23
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