Exercise 2.8

Question

Construct a Borel set $E \subset R^1$ such that
    \begin{equation*}
      0 < m(E \cap I) < m(I)
    \end{equation*}
    for every nonempty segment $I$. Is it possible to have $m(E) < \infty$ for
    such a set?

Amit Awasthi · Accepted Answer

\begin{proof}[Solution]
      Let $\{E_n\}_0^\infty$ be sets as constructed in Exercise 2.7
      translated onto the set $[n,n+1]$ and also copied to $[-n-1,-n]$, with
      parameter $\epsilon = 2^{-n-2}$. Then, $\bigcup_0^\infty E_n$ has measure $1$
      and intersects every segment in positive measure, but $\mu(E \cap I) <
      \mu(I)$ for any segment $I$ since every segment $I$ contains an interval
      which was removed in the construction of some $E_n$.
\end{proof}

Exercise 2.8

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Exercise 2.8

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