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Let $U\subset\mathbb{R}^n$ be an open set and $B\subset\mathbb{R}^n$ be an open ball. Let $D\subset B$ be dense in $B$. Also assume $D\subset U$. Can anything positive be said about $U$ "almost" containing $B$? I know that generally $B\subset U$ is not true.

But maybe it is true that $U$ always contains a set $T \subset B$ such that $|B\setminus T|=0$? Any other result is also appreciated. If it helps, assume that $U$ is bounded.

LordOfNumbers
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  • For a counterexample in the $n=1$ case, take the complement in $B$ of a "fat Cantor set": see e.g. this post. Easily adabpted to get counterexamples for general $n$. – Robert Israel Jan 18 '24 at 17:21
  • A good source on this is Oxtoby’s book Measure and category, Springer-Verlag. – Mittens Jan 18 '24 at 22:45
  • If $D$ is, say countable, (or more generally, any dense measure $0$ set in $B$) then an open subset of it can have arbitrarily small measure, by definition. – M W Jan 19 '24 at 03:36

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