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I'm doing some self educating about Measure Theory and I've come across two exercises I haven't managed to solve:

  1. Given $\mathcal{L}$ the Lebesgue measure on $\mathbb{R}$ construct a Borel-Set $\mathbb{E}\subseteq\mathbb{R}$ such that $$0<\mathcal{L}\left(E\cap\left[a,b\right]\right)<\mathcal{L}\left(\left[a,b\right]\right)$$ for all $-\infty<a<b<\infty$

  2. Show that if $A\subseteq\mathbb{R}$ is a measurable set for which there is an $\alpha\in\left[0,1\right)$ such that $$\mathcal{L}\left(A\cap\left[a,b\right]\right)\leq\alpha\mathcal{L}\left(\left[a,b\right]\right)$$ for all $\left[a,b\right]\subseteq \mathbb{R}$ then $\mathcal{L}\left(A\right)=0$

Help would be very appreciated as I am quite stumped.

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  • $\begingroup$ See here and here. $\endgroup$ – Umberto P. Nov 4 '13 at 19:14
  • $\begingroup$ Hmm, those are indeed the questions. I tried running a search but I didn't manage to phrase myself in a way that brought up those results... Thanks a lot :) $\endgroup$ – Serpahimz Nov 4 '13 at 19:18

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