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Let $O \subset \mathbb{R}^N$, $N \geq 3$, be a bounded set and $\mu$ the normalized probability Lebesgue measure on the induced $\sigma$-algebra of $O$. Therefore, $\mu(O) = 1$. Due to the regularity of $\mu$, we have

\begin{align} \forall\ \varepsilon > 0,\ \exists\ K = K(\varepsilon) \subset O,\ K \text{ compact}:\ \mu(K) \geq 1 - \varepsilon. \end{align}

Define $A := O \setminus K$ and let $d(\cdot,\cdot)$ be the Euclidean metric. Is the following statement true in general?

\begin{align} \forall\ \epsilon > 0,\ \forall\ x \in A,\ \exists\ y = y(\epsilon, x) \in K:\ d(x,y) < \epsilon. \end{align}

Remark: My intention with the above question is to get a better feeling of how metrically/topologically awkward a zero measure set, within a bounded set, can be.

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The set $A$ is open and so your statement is always false (because any point in $A$ is an interior point).

The answer is unrelated to the Lebesgue measure.

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