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This exercise appeared in my real analysis test last year, and is still puzzling me since then. Ironically, even the professor doubts if the b part is actually truth (still...)

Let $A \subset \mathbb{R}$

a) If $\displaystyle0, then for every $\alpha\in (0,1)$, exists an open interval $I$ such that $\displaystyle \alpha m^*(I) \leq m^*(A \cap I) $

b) If $ \displaystyle m^*(A \cap I) \leq \frac{m^*(I)}{2}$ for every open interval, then $m^*(A) = 0$

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    It seems it's true using the definition of the http://planetmath.org/LebesgueOuterMeasure.html (outer measure).2012-09-07
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    -1 to your professor2012-09-07
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    Notice that $a$ is essentially equivalent to $b$ by $x \Rightarrow y \sim \neg y \Rightarrow \neg x$.2012-09-07
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    Doesn't a) imply b) with $\alpha=\frac12$?2012-09-07
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    No. a) implies b) only if $m^*(A)<\infty$.2012-09-07
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    +1 for an interesting question...but -1 for a wholly unhelpful title.2012-09-07
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    @PeteL.Clark Does that mean you vote twice, or not at all?2012-09-07
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    @Graphth: Since it is not possible to vote twice, it must mean that I did not vote at all.2012-09-08
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    @PeteL.Clark What if I edit the title to make it meaningful? Will you then take back the -1 and leave only a +1?2012-09-08
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    @Graphth: Yes. In fact that was the intended meaning of my first comment. Sorry if I sacrificed meaning for snarky humor.2012-09-08
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    @PeteL.Clark No, it's fine. I'm just trying to be funny as well. I didn't even create the question, so I'm asking if "I" make the title better, will that affect your vote.2012-09-08

2 Answers 2

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Let $\varepsilon>0$ and $\{I_n\}_n$ a collection of open intervals which cover $A$ and such that $\sum_{j=1}^{+\infty}m^*(I_j)-m^*(A)\leq 2\varepsilon$. (the outer measure is finite). We have $$m^*(A)\leq \sum_{j=1}^{+\infty}m^*(A\cap I_j)\leq \frac 12\sum_{j=1}^{+\infty}m^*(I_j)\leq \frac{m^*(A)}2+\varepsilon,$$ hence $$0\leq \frac{m^*(A)}2\leq \varepsilon$$ and we are done.

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    Drat! you beat me. (+1)2012-09-07
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    How do you know that "the outer measure is finite"?2012-09-07
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    We work with $A_n:=A\cap [-n,n]$ to see that $m^*(A\cap [-n,n]=0$ (it's what you did in your answer).2012-09-07
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Consider $A_n:=A\cap\subseteq [-n,n]$ for $n\in \mathbb N$. Let $\mu:=m^*(A_n)$. Then $\mu\le 2n<\infty$. If $\varepsilon>0$, then there exists a countable collection of Intervals $I_k$, $k\in \mathbb N$, such that $A\subseteq\bigcup_k I_k$ and $\sum_k m^*(I_k) < \mu+\varepsilon$. We obtain $\mu \le \sum_k m^*(I_k\cap A_n) \le \sum_k \frac 12 m^*(I_k)=\frac 12\sum_k m^*(I_k)<\frac{\mu+\varepsilon}2$, hence $\mu<\varepsilon$. Since $\epsilon>0$ was arbitrary, we conclude $\mu=0$, i.e. $m^*(A_n)=0$.

Finally, $A=\bigcup_n A_n$ implies $m^*(A)\le \sum_n m^*(A_n)=0$.