Prove that $\lim_{m(I)\rightarrow 0}\frac{m(E\cap I)}{m(I)}=c$ for some $c$ and moving interval $I$.

Asked Apr 22 '19 at 06:35

Active May 14 '19 at 15:02

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Let $E$ be a set such that $m(E)>0$, $E \subset (0,1)$. Prove that there exists $c>0$ such that for some moving interval $I$, $$\lim_{m(I)\rightarrow 0}\frac{m(E\cap I)}{m(I)}=c$$

Proof: $m(E)=1$

My attempt:

I have proof E is dense in(0,1),proof by contradiction if E nowhere dense in (0,1), there exist $(a,b)$ such that $(a,b)\subset (0,1)-E$, then $m((a,b)\cap E)=0$,we get a conflict

I think Maybe need the theorem using in construct unmeasurable set: If m(E)>0, then for any $a \in (0,1)$,there exist an interval J,such that $m(E\cap J)>am(J)$

edited May 14 '19 at 15:02

Lev Bahn

2,892

asked Apr 22 '19 at 06:35

jackson

3

$E$ being dense does not tell you that $m(E)=1$. – nicomezi Apr 22 '19 at 07:59
@nicomezi you are right – jackson Apr 22 '19 at 08:16
1

Do not add back irrelevant tags. Thank you. – Asaf Karagila Apr 22 '19 at 12:20
@AsafKaragila sorry – jackson Apr 22 '19 at 12:23
I think getting contradiction from assuming that $E$ is nowhere dense does not tell you $E$ is dense since being not dense does not mean it is nowhere dense – Lev Bahn May 14 '19 at 13:59
@LevBan yes but I don’t know how to solve the question can you help me..I think maybe use contradiction? – jackson May 14 '19 at 14:01
1

I am not sure but it would be helpful? https://math.stackexchange.com/questions/2896620/let-lambdaa-be-the-lebesgue-measure-of-a-there-exists-an-interval-i-su/2896665#2896665 – Lev Bahn May 14 '19 at 14:02
@LevBan thanks I am reading it – jackson May 14 '19 at 14:04
I still can’t work through it.. can you help me.. – jackson May 14 '19 at 14:18
@LevBan I still can’t work through it can you help me – jackson May 14 '19 at 14:18
@jackson I am not sure if I have an idea, but I will try. I will write the answer as soon as I can come up with the solution :) – Lev Bahn May 14 '19 at 14:22
@LevBan wow thanks a lot – jackson May 14 '19 at 14:22
@jackson there is a proof for your problem. There are already several different proof for this problem :) hope it helps.
https://math.stackexchange.com/questions/596676/lebesgue-density-theorem-in-the-line
– Lev Bahn May 14 '19 at 14:46
Thanks I am reading it – jackson May 14 '19 at 14:53
@jackson great! – Lev Bahn May 14 '19 at 15:28

Prove that $\lim_{m(I)\rightarrow 0}\frac{m(E\cap I)}{m(I)}=c$ for some $c$ and moving interval $I$.

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