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Is there a measurable set $A$ such that $m(A \cap B) = \frac12 m(B)$ for every open set $B$?

Is there a measurable set $E \subset [0,1]$ such that for any $0 < a < b<1$, the Lebesgue measure $$m(E \cap [a,b])= \frac{b-a}{2}?$$

I am stumped and have no idea.

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Is this a homework problem? If so, please add the homework tag. In any case, what have you already tried? –  Adam Saltz Dec 8 '11 at 0:07
    
I am learning real analysis from an old grad level book I got, and this is just one of the problems in it. So maybe a homework problem? –  Alex J. Dec 8 '11 at 0:14
    
To make the linked question more directly applicable, note that an open subset $U$ of $[0,1]$ is a countable union of disjoint open intervals, and it follows that $m(E\cap U)=\frac{1}{2}\cdot m(U)$. (And then you can take $U$ such that $E\subset U$ and $m(U)<1$ to get a contradiction.) –  Jonas Meyer Dec 8 '11 at 6:27
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marked as duplicate by GEdgar, JavaMan, t.b., Zhen Lin, Dylan Moreland Dec 8 '11 at 6:18

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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No, there is no such a set.

Since $E$ is measurable (with positive measure) for all $\epsilon > 0$ there exist a sequence of intervals $(I_n)$ in $[0,1]$ such that $E \subset \bigcup I_n$ and $$ \sum m(I_n) < m(E) + \epsilon. $$ Hence we obtain that $$ \frac{3}{4} \sum m(I_n) < m(E). $$ Thus $$ \frac{3}{4} \sum m(I_n) < m(E) \leq m(E \cap \bigcup I_n) = m(\bigcup E \cap I_n) \leq \sum m(E \cap I_n), $$ which implies that there exists positive integer $k$ such that $$ \frac{3}{4} m(I_k) \leq m(E \cap I_k). $$ The last inequality contradicts your assumption that for any interval $I$ we have $m(E \cap I) = m(I)/2$.


Whats more, in the exactly same way you can prove general fact:

If $E \subset \mathbb{R}$ is a measurable set with positive measure and $0 \leq c < 1$ than there exists an interval $I$ such that $$ m(E \cap I) \geq c \cdot m(I). $$

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I changed some $A$s to $E$s. Hope I caught all of them! –  Dylan Moreland Dec 8 '11 at 1:09
    
@DylanMoreland: Thank you very much! There was one more, in the last line :). –  xen Dec 8 '11 at 17:18
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No there can be no such $E$.

Let $\chi_E$ be the characteristic function of $E$. At every Lebesgue point $x$ of $\chi_E$, we have

$$\chi_E(x) = \lim_{r\to 0}\ \frac{1}{m(B_r)} \int_{B_r(x)} \chi_E(y) \ dy$$

Since almost every point of an integrable function is a Lebesgue point, the set of $x$ for which this limit is not equal to $0$ or $1$ must have measure $0$.

In particular, it cannot be true that

$$m(E\cap[a,b]) = \frac{b-a}{2}$$

for all $a,b$, since that would imply that also

$$\frac{1}{2} = \frac{m(E\cap[x-r,x+r])}{m([x-r,x+r])} = \frac{1}{m(B_r)} \int_{B_r(x)} \chi_E(y) \ dy$$

for all $x$ and all small enough $r>0$. (Which would be in contradiction to the fact about Lebesgue points above.)

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