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Let $E$ be a measurable subset of $[0,1]$. Then we know that we can choose an open set $G$ and a closed set $F$ such that $F\subset E \subset G \subset [0,1]$. For each $t\in [0,1]$, define

$\delta(t) = \begin{cases} \text{dist}(t,G^c),&\text{if $t\in F$}\\ \text{min}\{\text{dist}(t,b(G)),\text{dist}(t,F)\},&\text{if $t\in G\smallsetminus F$}\\ \text{dist}(t,F),&\text{if $t\in [0,1]\smallsetminus G$.} \end{cases} $

Here we use the notation $b(G)$ to mean the boundary of $G$. Because the sets $G^c$, $b(G)$, and $F$ are closed, it follows that $\delta(t)>0$ for each $t\in [0,1].$ This defines a positive function $\delta$ defined on $[0,1].$ Cousin's Lemma therefore assures that a division $D=\{(t_i,I_i)\}_{i=1}^{n}$ exists such that for each $i=1,\dots,n$ we have $t_i\in [0,1]$ and

$I_i\subset (t_i-\delta(t_i),t_i+\delta(t_i)).$

Some authors call $D$ as a $\delta$-fined free tagged division of $[0,1].$

And here is my question:

How do we show (if it is true) that

$(D)\sum_{t_i\in E}[\mu(I_i)-\mu(E\cap I_i)] \leq \mu(G\smallsetminus F)$ and

$(D)\sum_{t_i\notin E}\mu(E\cap I_i)] \leq \mu(G\smallsetminus F)$

where $\mu$ denotes the Lebesgue measure.

Any tips??Thanks in advance...

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up vote 2 down vote accepted

The choice of $\delta$ implies the following two properties of the tagged partition $(t_i,I_i)$: $$t_i\in G\implies I_i\subseteq G \tag{1}$$ $$t_i\notin F\implies I_i\cap F=\varnothing \tag{2}$$

Since the intervals form a partition of $[0,1]$, property (1) implies $$\sum_{t_i\in G} \mu(I_i) \le \mu(G)\tag{3}$$ and property (2) implies $$\sum_{t_i\in G} \mu(F\cap I_i) = \mu(F)\tag{4}$$ Therefore, $$\sum_{t_i\in G} (\mu(I_i) - \mu(F\cap I_i)) \le \mu(G\setminus F)\tag{5}$$ which is stronger than the your first inequality.

If $t_i\notin F$, then by (2) $E\cap I_i\subseteq G\setminus F$. Therefore, $$\sum_{t_i\notin F} \mu(E\cap I_i)\le \mu(G\setminus F)\tag{6}$$ which is stronger than your second inequality.

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I still have to verify(if I can do it)the 6 inequalities.:)Much better if you can add some details. Anyway, many many thanks to you. I have already an answer to the problem but it seems that your answer is simpler and nice. – juniven Jan 28 '13 at 4:03
@jum But if I add more details, they will involve more inequalities, which will also need to be verified. That will call for more inequalities, and so on to infinity... – user53153 Jan 28 '13 at 4:16
I understand. Do not be bothered. The suggested inequalities are already enough. – juniven Jan 28 '13 at 4:20
As of this time, I can follow everything except (4). Maybe later I can fix it. I take a break first.:) – juniven Jan 28 '13 at 5:25
@jun I made a mistake in (4) and (5): edited now. Let me know if anything else is unclear. – user53153 Jan 28 '13 at 5:29

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