Construction Of A Cantor-like set with certain property

Asked Mar 28 '15 at 00:13

Active Mar 28 '15 at 01:39

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Is there a way to construct a measurable set $E \subset [0,1]$ with the property that for every interval $[a,b] \subset [0,1]$, both $[a,b]\cap E $ and $[a,b]$\ $E$ have positive Lebesgue measure?

I think you have to construct a cantor set of positive measure and add in each of the intervals that are ommited another cantor like set with positive measure, and continue this procedure indefinitely but I cannot rigorously prove that its correct. Any ideas? Thanks in advance.

edited Mar 28 '15 at 01:23

asked Mar 28 '15 at 00:13

SpawnKilleR

bof means Spawn should show his own work before commanding us to do this for him. – GEdgar Mar 28 '15 at 00:19
$( [a,b] \cap E) \cup ( [a,b] \setminus E) = [a,b]$ – Yuugi Mar 28 '15 at 00:34
This has to be for every interval [a,b] not just a certain choice. – SpawnKilleR Mar 28 '15 at 00:38
I know it is for every interval, I am just giving you an idea. Also, maybe look at the Fat-Cantor Set. – Yuugi Mar 28 '15 at 00:38
Sorry, I was wrong. Not every interval will intersect with positive measure. – Yuugi Mar 28 '15 at 00:57

Construction Of A Cantor-like set with certain property

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