Limit of decreasing sequence of open sets

lebesgue-measuremeasure-theoryreal-analysis

There two parts to the problem for a compact set $E$:

  1. Define
    $$
    \mathcal{O}_n = \{x \in E: d(x, E) < 1/n \}
    $$
    show that the $m(E) = \lim_{n \rightarrow \infty} m(\mathcal{O}_n)$, where $m$ is the standard Lesbegue measure.
  2. Show that the above assertion maybe false if (a) $E$ is closed but unbounded and (b) $E$ is open but bounded.

I have proved all parts except for 2(b). For 2(a) i take $E = \{1,2,3, … \}$. For 2(b) I am at a loss. Any pointers would help.

Best Answer

Let $\{q_k\}$ be an enumeration of $[0,1]\cap \mathbb Q$ and define

$$ E = \bigcup_{k\in \mathbb N} ( q_k -2^{-k-2}, q_k+ 2^{-k-2}).$$

Then $m(E) \le 1/2$ and $E$ is open and bounded. But for all $n\in \mathbb N$, since $E$ is dense in $[0,1]$, we have $[0,1] \subset \mathcal O_n$. Thus $m(\mathcal O_n) \ge 1$.

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