# Construction of a Borel set

I have been trying to solve this question but reaching nowhere

Starting from a countable basis of $$\mathbb R$$ ,I am asked to construct a Borel set such that $$0 for every non empty segment I.

And then must $$E$$ be of infinite measure?

Here $$m$$ denotes the Lebesgue measure

• such that what? – Marios Gretsas Oct 3 at 11:54

Hint: for a basis, take the intervals $$(a,b)$$ where $$a < b$$ are rational.
For your set $$E$$, take a union of "fat Cantor sets", one for each of these intervals.
• Is it obvious that $m(E\cap I)<m(I)$ for all $I$? @RobertIsrael – Kabo Murphy Oct 3 at 12:18
Say $$(r_j)$$ is a dense sequence. Choose $$a_j>0$$ so that $$a_k>\sum_{j=k+1}^\infty a_j,$$for example $$a_j=1/3^j$$, then define $$I_j=(r_j-a_j,r_j+a_j),$$ $$E_k=I_k\setminus\bigcup_{j=k+1}^\infty I_j$$and $$E=\bigcup E_k.$$