Exercise 2.7

Question

If $0 < \epsilon < 1$, construct an open set $E \subset [0,1]$ which is dense in $[0,1]$, such that $m(E) = \epsilon$. (To say that $A$ is dense in $B$ means that the closure of $A$ contains $B$).

Amit Awasthi · Accepted Answer

\begin{proof}[Solution]
      We claim the complement $E$ of $K$ we constructed in Exercise 2.6 suffices. We have $\mu(E) = \epsilon$ since $\mu(K) = 1 - \epsilon$. On
      the other hand, $\overline{E} = [0,1]$ since $K$ has empty interior by
      Exercise 2.6.
\end{proof}

Exercise 2.7

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Exercise 2.7

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