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Is that correct that, for any open set $S \subset \mathbb{R}^n$, there exists an open set $D$ such that $S \subset D$ and $D \setminus S$ has measure zero?
I think it is correct and I guess I have seen the proof somewhere before, but I cannot find it in any of my books, if it is wrong, please give me a counter-example.
Also is the same correct for a closed set such that it's interior is not of measure zero?

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This is false if for example $S$ is an open ball. –  Chris Eagle Nov 23 '12 at 14:50
    
(I assume that "$\subset$" means "is a proper subset of" and "$D - S$" means "$D \setminus S$", though both of these usages are nonstandard) –  Chris Eagle Nov 23 '12 at 14:52
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@Chris: Yes, I realized that just after I posted. (It’s actually my preferred usage, but I don’t see it often enough to expect it!) I disagree about non-standard: the first is perfectly standard, and the second is merely old-fashioned. –  Brian M. Scott Nov 23 '12 at 14:52
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What about everyone's favourite open set: $\varnothing$? –  Arthur Fischer Nov 23 '12 at 15:15
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I believe what you may be thinking of are certain regularity conditions on Lebesgue measure. They are: if $E$ is a Lebesgue measurable set, then for every $\epsilon > 0$: 1) There is an open set $U$ containing $E$ with $|U \setminus E| < \epsilon$ 2) There is a closed set $F$ inside of $E$ with $|E\setminus F| < \epsilon$ 3) If $|E| < \infty$, then there is a finite union of closed cubes so that the symmetric difference with $E$ has measure less than/equal to $\epsilon$. –  anonymous Nov 23 '12 at 16:17

1 Answer 1

Assume that $S\subset D$ with $S,D$ open and $\mu(D\setminus S)=0$. If $D$ is not contained in $\overline S$, then the nonempty open set $D\setminus \overline S$ has positive measure. Therefore, $D\subseteq \overline S$. Therefore any open set $S$ with the property that $$\tag1\partial S\subseteq \partial(\mathbb R^n\setminus \overline S)$$ is a counterexample to your conjecture: Any open ball araound a point in $\partial S$ then intersects $\mathbb R^n\setminus \overline S$ in a nonempty open set of positive measure. For example, an open ball or virtually any open set with a smooth boundary has property (1).

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