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Construct a Lebesgue measurable subset A of $\mathbb{R}$ so that for all reals $a<b$, $$0<m(A\cap(a,b))<b-a $$ under the usual Lebesgue measure $m$.

And then show that if $m(A\cap(a,b))\leq\frac{b-a}{2}$ for any $a<b\in\mathbb{R}$, then $m(A)=0$.

The answer to the first part is a dublicate, but as far as the second bit I am feeling stumped. Any thoughts? Thank you for the help.

wfw
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    I do not see how the first and the second are consistent. Suppose you had an example of $A$ from the first statement. Then the second statement would imply $m(A\cap(a,b))\gt \frac{b-a}{2}$ for all $a \lt b$, so $m(A^c\cap(a,b))\lt \frac{b-a}{2}$ for all $a \lt b$ so $m(A^c)=0$ so $m(A\cap(a,b))= b-a$ for all $a \lt b$, contrary to the first statement. (Perhaps it depends on whether the second statement is any or all) – Henry Aug 29 '17 at 06:51
  • I thought for a second I typed it up wrong but thats how it is written. I think the writer meant that for all $a<b\in\mathbb{R}$ – wfw Aug 29 '17 at 07:00
  • The answer might go over my head, but: what's the difference, if any, between a Borel set and Lebesgue set? – Robert Soupe Aug 31 '17 at 06:37

1 Answers1

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The second part is immediate from Lebesgue's Theorem. For the first part, you have to construct a sequence of Cantor like sets of positive measure and take their union. After constructing one Cantor set of positive measure do the same in each of the intervals removed, and so on. The union of all the Cantor sets has the desired properties.

Kavi Rama Murthy
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  • In the statement of the theorem in "Lebesgue's Density Theorem" in Wikipedia, an $\in$-nbhd of a point $p$ should be a ball (or interval) centered at $p$, of radius $\in.$ – DanielWainfleet Aug 29 '17 at 16:38