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Comparing the Lebesgue measure of an open set and its closure

Given an open set in $\Bbb R^2$ of finite Lebesgue measure, why does the boundary of its closure have measure zero?

  • 4
    This is false, isn't it? Consider the inside of a positive-measure Jordan curve.2012-08-24
  • 4
    You can take a look at here : http://mathoverflow.net/questions/25993/sets-with-positive-lebesgue-measure-boundary.2012-08-24
  • 1
    Or [here](http://math.stackexchange.com/a/185950/6179).2012-08-24

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