Is there a Lebesgue measurable subset $A \subset R$ such that for every interval $(a,b)$ we have $0 < \lambda(A\cap(a,b))< (b-a)$

Asked May 06 '19 at 23:21

Active May 11 '19 at 11:52

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Question: Is there a Lebesgue measurable subset $A \subset R$ such that for every interval $(a,b)$ we have $$0 < \lambda(A\cap(a,b))< (b-a)$$

It looks like the answer is no, I'm trying to use the Lebesgue differentiation theorem on A in order to do so.

Also, If we assume there's a set $A$, $\forall (a,b) \subset (0,1)$, $0 < \lambda(A\cap(a,b))< (b-a)$, Does there exist an open interval $I_n \subset (0,1)$ that will hold as a contradiction?

edited May 11 '19 at 09:32

mathworker21

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asked May 06 '19 at 23:21

Ilan Aizelman WS

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See Problem #8 in http://www.math.utah.edu/~ptrapa/6210/ps2-solutions.pdf. – copper.hat May 09 '19 at 19:50

Is there a Lebesgue measurable subset $A \subset R$ such that for every interval $(a,b)$ we have $0 < \lambda(A\cap(a,b))< (b-a)$

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